Simultaneous extreme values of zeta and L-functions

Abstract

We show that distinct primitive L-functions can achieve extreme values simultaneously on the critical line. Our proof uses a modification of the resonance method and can be applied to establish simultaneous extreme central values of L-functions in families.

1 Introduction

Extreme values of L-functions have attracted considerable attention in recent years. This surge of activity is largely due to the introduction of the resonance method of Soundararajan [44] and its subsequent developments. This is a versatile method which allows one to show that for various families of L-functions ${\mathcal {F}}$ with analytic conductor C,

$$\begin{aligned} \max _{\begin{array}{c} \pi \in {\mathcal {F}}\\ {\text {cond}}\pi \asymp C \end{array}}\left| L\left( \tfrac{1}{2},\pi \right) \right| \geqslant \exp \left( c\sqrt{\frac{\log C}{\log \log C}}\right) \end{aligned}$$

(1.1)

for some constant $c=c({\mathcal {F}})>0$. In the case of the Riemann zeta function, Bondarenko–Seip [8] recently made the significant improvement

$$\begin{aligned} \max _{t\in [0, T]}\left| \zeta \left( \tfrac{1}{2}+it\right) \right| \geqslant \exp \left( c '\sqrt{ \frac{\log T\log \log \log T}{\log \log T}}\right) . \end{aligned}$$

(1.2)

Their modification of the resonance method can be extended to other families [12], although a severe restriction is that the L-functions must have non-negative coefficients.¹

Extreme values of products of L-functions are also possible and in particular the bound (1.1) holds when $L(s,\pi )$ factorises. For example, we know [5, Theorem 1.12] that for fixed cusp forms f, g there exists a non-trivial primitive character $\chi$ modulo a prime q such that

$$\begin{aligned} |L(1/2, f \otimes \chi )L(1/2, g\otimes \chi )|\geqslant \exp \left( c_{f, g}\sqrt{\frac{\log q}{\log \log q}}\right) . \end{aligned}$$

(1.3)

In t-aspect, Aistleitner–Pańkowski [1] showed that (1.1) holds for non-primitive L-functions in the Selberg class, whereas bounds of the strength (1.2) can be demonstrated for Dedekind zeta functions [7]. These results contain an interesting feature: when $L(s,\pi )$ is a product of L-functions one can achieve a larger constant c. Precisely, the t-aspect results show that for a product of m distinct primitive L-functions $c=\sqrt{m}$ is admissible, whereas for individual L-functions $c=1$.

As pointed out in [7], this presents a dichotomy. Either extreme values of L-functions are occurring simultaneously, or one can demonstrate even larger values for an individual L-function. The question of whether L-functions can attain extreme values simultaneously is natural and forms the focus of this paper. Another motivation for the present work is that extreme values provide a testing ground for the statistical independence of L-functions; a property which, if held at the extremes, can have deep arithmetic consequences.²

Previous results on joint extreme values have been obtained when we move away from the central point. For fixed $1/2<\sigma <1$, under suitable assumptions on the L-functions Mahatab–Pańkowski–Vatwani [38] used Diophantine approximations to show the existence of large values of $L_j(\sigma +it_j)$ with the $t_j$’s living in a small neighbourhood of each other. By considering the joint value distributions of $L(\sigma +it, \chi _j)$, Inoue–Li [25] recently showed the existence of simultaneous large values of Dirichlet L-functions in the strip $1/2< \sigma <1$.

Less is known for joint extreme values at the central point. It was observed by Selberg [42] and later proved by Bombieri–Hejhal [6] that the joint distribution of L-functions (satisfying suitable assumptions) splits as the product of the distributions:

$$\begin{aligned}&\frac{1}{T}\textrm{meas}\bigg \{t\in [T,2T]:\frac{\log \left| L_j\left( \tfrac{1}{2} +it\right) \right| }{\sqrt{\tfrac{1}{2}\log \log T}}\geqslant V_j,\,\, j=1,\ldots , m\bigg \} \nonumber \\&\quad \sim \prod _{j=1}^m \int _{V_j}^\infty e^{-x^2/2}\frac{dx}{\sqrt{2\pi }} \end{aligned}$$

(1.4)

as $T\rightarrow \infty$ for any fixed $V_j$. Recently Inoue–Li [24] extended the range of $V_j$ in (1.4) to $V_j\ll (\log \log T)^{1/10}$ which yields simultaneous large values of size $\exp \big ( c(\log \log T)^{3/5-\epsilon }\big )$. Simultaneous extreme values of size (1.1) would be desirable, but they are harder to detect using joint distributions since these events are very rare. Indeed, even bounds for the joint distribution in the range $V_j\approx \sqrt{\log \log T}$ are not available unconditionally due to our lack of knowledge on mixed moments (although the conjectured asymptotics [19, 39] could likely be established up to order conditionally [17, 20]). Here, we utilise a modification of the resonance method to detect simultaneous large values of central values of L-functions, overcoming inefficiencies from methods of moments and Diophantine approximation. Our first result gives simultaneous large values of two L-functions on the critical line.

Theorem 1

Let $L_1(s), L_2(s)$ be either the Riemann zeta function, a primitive Dirichlet L-function or the L-function attached to a primitive cusp (holomorphic or Maaß) form on GL(2) over ${\mathbb {Q}}$. Then there exists some positive constant c depending on $L_1, L_2$ such that for sufficiently large T, we have

$$\begin{aligned} \max _{t\in [T,2T]}\min \left( \left| L_1\left( \tfrac{1}{2}+it\right) \right| , \left| L_2\left( \tfrac{1}{2}+it\right) \right| \right) \geqslant \exp \bigg (c\sqrt{\frac{\log T}{\log \log T}}\bigg ). \end{aligned}$$

Remark 1

The constant depends on the degree of the L-functions in question. If $L_1, L_2$ are both Dirichlet L-functions, then we can take $c=\sqrt{17/66}+o(1)$ and if at least one of $L_1$ and $L_2$ is a GL(2) L-function, we can take $c=\sqrt{ {(1-2\theta )}/{12}}+o(1)$ where $\theta$ is an admissible bound towards the Ramanujan conjecture for GL(2) over ${\mathbb {Q}}$. By work of Kim–Sarnak [28, Appendix 2], we can take $\theta =7/64$. For comparison, we mention that one can take $c=\sqrt{2}+o(1)$ for the product of $L_1L_2(1/2+it)$ (see [1]).

Our method extends to other families and we shall describe a general principle below. For now, we illustrate this by demonstrating simultaneous large central values of twists of GL(2) cusp forms, refining (1.3).

Theorem 2

Let f, g be fixed primitive cusp forms of level $r, r'$ and trivial central character. There exists a positive constant c depending only on f, g such that for all primes q sufficiently large in terms of f, g, there exists a non-trivial character $\chi \bmod q$ such that

$$\begin{aligned} \min \Big (|L(1/2, f\otimes \chi )|, |L(1/2, g\otimes \chi )|\Big )\geqslant \exp \Big (c \sqrt{\frac{\log q}{\log \log q}}\Big ). \end{aligned}$$

Remark 2

We give an explicit constant c in terms of f, g in the proof. In a generic situation, we can take $c={1}/{12\sqrt{10}}+o(1)$, which is half of the constant $c_{f, g}={1}/{6\sqrt{10}}+o(1)$ for the product in (1.3) [5, Remarks 7.3, 7.20], as one would expect.

We now describe our method. The idea is to use a resonator that picks out a large value of the product of the L-functions that, at the same time, is significantly bigger than their sum, thus giving simultaneous large values. We detail this in the t-aspect, although the principle extends more generally. Our aim is to find a Dirichlet polynomial R(t) such that for large V,

$$\begin{aligned}{} & {} \int _T^{2T}\left( \left| L_1\left( \tfrac{1}{2}+it\right) L_2\left( \tfrac{1}{2}+it\right) \right| ^2-V\left| L_1 \left( \tfrac{1}{2}+it\right) \right| ^2-V\left| L_2\left( \tfrac{1}{2}+it\right) \right| ^2\right) |R(t)|^2dt\nonumber \\{} & {} \quad >0. \end{aligned}$$

(1.5)

If this holds then there exists a $t\in [T,2T]$ such that

$$\begin{aligned} \left| L_1\left( \tfrac{1}{2}+it\right) L_2\left( \tfrac{1}{2}+it\right) \right| ^2 -V\left| L_1\left( \tfrac{1}{2}+it\right) \right| ^2-V\left| L_2\left( \tfrac{1}{2}+it\right) \right| ^2>0 \end{aligned}$$

which implies that both $|L_1(1/2+it)|^2,|L_2(1/2+it)|^2>V$. We choose R(t) to pick out large values of the product and once the asymptotic formulae for twisted second moments in (1.5) have been established, we can find the desired size for V. This approach uses the larger values of the product in a key way, but also includes the required upper bound information (as is necessary to rule out excessively large values of individual L-functions).

For multiple L-functions we can aim to find a value of t for which

$$\begin{aligned} \prod _{j=1}^m \left| L_j\left( \tfrac{1}{2}+it\right) \right| ^2-V\sum _{1\leqslant i\leqslant m}\prod _{\begin{array}{c} j=1\\ j\ne i \end{array}}^m\left| L_j\left( \tfrac{1}{2}+it\right) \right| ^2 >0. \end{aligned}$$

(1.6)

Unfortunately, asymptotic formulae for the twisted second moments of multiple or higher degree L-functions are currently out of reach. However, asymptotics are not strictly necessary since reasonably sharp bounds would suffice. A lower bound for the first term on the left of (1.6) can be achieved fairly easily through the Cauchy–Schwarz inequality. This allows one to replace the second moment with a first moment which is more tractable. In fact, this first moment can be computed for any number of L-functions.

To obtain upper bounds for the second term in (1.6) we note that there exist several instances in the literature [11, 14, 32] where, if one doesn’t have access to asymptotics for twisted second moments, a sharp upper bound can still be achieved on the Riemann Hypothesis by applying the methods of Harper [17]. As it stands, Harper’s method is designed for a fixed 2k-th moment of an L-function and thus sensitive to values of size $(\log T)^k$. However, with some modifications, in particular by focusing on very small primes, these methods can be made suitable for extreme values. With this in hand we can exhibit simultaneous large values for many L-functions in higher degrees under the Riemann Hypothesis for these L-functions.

Theorem 3

Let $\pi _j$, $j=1,\cdots , m$ be irreducible unitary cuspidal automorphic representations of $GL(d_j)$ over ${\mathbb {Q}}$ such that $\pi _i\not \cong \pi _j$ for $i\not =j$. Assume the generalised Riemann Hypothesis for all $L(s, \pi _j)$ $j=1, \dots , m$ and assume the generalised Ramanujan conjecture for $L(s, \pi _j)$ if $d_j\geqslant 3$. Then for sufficiently large T we have

$$\begin{aligned} \max _{t\in [T,2T]}\min \left( \left| L\left( \tfrac{1}{2}+it,\pi _1\right) \right| ,\ldots ,\left| L\left( \tfrac{1}{2}+it,\pi _m\right) \right| \right) \geqslant \exp \bigg (c\sqrt{\frac{\log T}{\log \log T}}\bigg ) \end{aligned}$$

for any positive constant $c< \frac{1}{\sqrt{2m}}$.

Additional work is required to remove the generalised Ramanujan conjecture in the case of Maaß forms. Throughout most of the proof we can work under weaker assumptions, in particular when computing the mean values. However in the final step when establishing extreme values more strict control on the size of $a_\pi (p)$ is required. One sufficient condition is a Mertens’ type estimate for the fourth power moment:

$$\begin{aligned} \sum _{p\leqslant X}\frac{|a_\pi (p)|^4}{p}\ll \log \log X. \end{aligned}$$

Therefore, the assumption of the generalised Ramanujan conjecture can be avoided for GL(2) Maaß forms by the functoriality of symmetric powers established by Kim [27] (and also for self-dual GL(3) L-functions using Gelbart–Jacquet [15] and Kim [27], although we do not state this in Theorem 3 for concision).

We also remark that Theorem 3 can be proved unconditionally for three distinct Dirichlet L-functions. Here, the upper bounds for the twisted second moments of $L(1/2+it,\chi _i)L(1/2+it,\chi _j)$ are (essentially) available being generalisations of the twisted fourth moment of the Riemann zeta function [4, 18, 21].

The versatility of the resonance method in families of L-functions extends to simultaneous values, both large and small. Our methods allow for a general principle which we now describe. Let ${\Pi }$ be some family and let $\{L(s,f_j)\}_{j=1}^m$ be fixed L-functions. Suppose we can lower bound

$$\begin{aligned} \sum _{\pi \in \Pi } \prod _{j=1}^mL(1/2, f_j\otimes \pi )|R(\pi )|^2 \end{aligned}$$

(1.7)

and upper bound

$$\begin{aligned} \sum _{\pi \in \Pi } \prod _{j\ne i}|L(1/2, f_j\otimes \pi )|^2|R(\pi )|^2 \end{aligned}$$

(1.8)

in a reasonably sharp way. Then via the inequality (1.6) one can show the existence of $\pi \in \Pi$ such that $|L(1/2,f_j\otimes \pi )|$ are large simultaneously. Here, one uses the Cauchy–Schwarz inequality to get a lower bound for the mixed absolute second moment using (1.7) together with an estimate for $\sum _{\pi \in \Pi }|R(\pi )|^2$. If upper bounds for (1.8) are not immediately accessible, then one can apply our adaption of Harper’s methods (Sect. 6) to give conditional results.

The situation for simultaneous small values is somewhat simpler. Here, we just require upper bounds for the sum

$$\begin{aligned} \sum _{\pi \in \Pi } \sum _{j=1}^m|L(1/2, f_j\otimes \pi )|^2|R(\pi )|^2 \end{aligned}$$

(1.9)

along with the simple fact that for non-negative a, b, the inequality $a+b\leqslant V$ implies $a,b\leqslant V$. In both the large and small value cases, the resonator should be chosen to pick out extreme values of the full product of L-functions.

We illustrate this principle in two families. As we have seen already in Theorem 2, it can be applied to give simultaneous large central values of L-functions of GL(2) cusp forms twisted by Dirichlet characters modulo q unconditionally, since the second moment theory has been well developed [5].

For simultaneous small values, we consider the family of holomorphic cusp forms twisted by quadratic characters $\chi _d(\cdot )=(\frac{d}{\cdot })$. Here, the mixed first moment

$$\begin{aligned} \sum _{\begin{array}{c} 0<d\leqslant X \end{array}}L(1/2,f\otimes \chi _{d})L(1/2,g\otimes \chi _{d}) \end{aligned}$$

is still unknown. Consequently, the simultaneous non-vanishing of quadratic twists of cusp forms remains an open question. (It is known from Munshi [40] that if there exists one quadratic character with simultaneous non-vanishing, then in fact there must be infinitely many such primitive quadratic characters.) Nevertheless, we can show that there are infinitely many d such that $L(1/2,f\otimes \chi _{d})$ and $L(1/2,g\otimes \chi _{d})$ get very small simultaneously.

Theorem 4

Let f, g be holomorphic cusp forms of weight $\kappa \equiv 0 \bmod 4$ for $SL_2({\mathbb {Z}})$ and let $\chi _{d}(\cdot )=(\frac{d}{\cdot })$ be the Kronecker symbol. Then for large X there exists $X\leqslant d\leqslant 2X$ and some $c>0$ such that

$$\begin{aligned} \max (L(1/2, f\otimes \chi _{d}),L(1/2,g\otimes \chi _{d}))\ll \exp \bigg (-c\sqrt{\frac{\log X}{\log \log X}}\bigg ). \end{aligned}$$

We remark that this result is unconditional since we can avoid the absolute second moments in (1.9) (which are currently out of reach, though significant progress has been made recently by Li [33]) and work directly with $L(1/2, f\otimes \chi _{d})$ as we already have non-negativity: $L(1/2, f\otimes \chi _{d})\geqslant 0$. This surprising fact is known unconditionally from the formula of Waldspurger [46] (see also [29]). In generic situations, one can take $c={1}/{\sqrt{5}}+o(1)$.

There are several other possibilities for simultaneous extreme values in families of L-functions. Examples of significant arithmetic interest are given by the families

$$\begin{aligned} \{L(1/2,f), L(1/2,f\otimes \chi _D): f\in {\mathcal {F}} \} \end{aligned}$$

where $\chi _D$ is a fixed quadratic character and $\mathcal {F}$ is either the family of Hecke eigencuspforms of even weight k for the full modular group with k tending to infinity, or the family of holomorphic newforms of fixed even weight k for the congruence subgroup $\Gamma _0(N)$ with N tending to infinity. In both the weight and (squarefree) level aspects, the required second moment formulae can be computed using Petersson’s formula (see [13, 23, 30, 44] for example). We also mention that for a large prime q, Dirichlet characters $\omega _1, \omega _2$ modulo q and f a Hecke eigenform for $SL_2(\mathbb {Z})$ (holomorphic or Maaß ), the simultaneous extreme values for the families

$$\begin{aligned}&\big \{L(1/2, \chi ), L(1/2, \omega _1\chi ), L(1/2, \omega _2\chi ): \chi \bmod q\big \}\\&\big \{L(1/2, f\otimes \chi ), L(1/2, \chi ): \chi \bmod q\big \} \end{aligned}$$

could be established using work of Zacharias [48].

We close this introduction with a few remarks on similarities with previous works and on the difficulties in extending our results to the strength of (1.2). We note that our proof utilises some control on both upper and lower bounds for L-functions. A similar idea has appeared in the recent work of Gun–Kohnen–Soundararajan [16] where they demonstrated large central values of linear combinations of L-functions by making one L-function large and at the same time keeping all other L-functions smaller. In contrast, we exhibit simultaneous extreme central values in families of L-functions where all of the L-functions attain large or small values.

A very natural question is whether one can attain simultaneous values of the strength of Bondarenko–Seip [8] given in (1.2). A key source of their improvement was the use of a resonator with support in numbers that are much bigger than T, although this results in considerable difficulties. First, to lower bound

$$\begin{aligned} \int _1^{T}L_1\left( \tfrac{1}{2}+it\right) L_2\left( \tfrac{1}{2}+it\right) |R(t)|^2dt \end{aligned}$$

with such a resonator we can require that the coefficients of $L_1(s)L_2(s)$ be positive after inserting a smooth weight with positive transform, as in [8]. This could be resolved with Dedekind zeta functions, for example. However, finding an upper bound for

$$\begin{aligned} \int _1^T\left| L_j\left( \tfrac{1}{2}+it\right) \right| ^2 |R(t)|^2dt \end{aligned}$$

when R is a long resonator seems a more substantial obstacle. In [8], to get an upper bound for $\int |R(t)|^2dt$, the resonator was chosen to have well-spaced phases but unfortunately it is not clear how such an R interacts with $|L_j(1/2+it)|^2$. If this issue could be overcome then one could deduce bounds of the form (1.2) for Dirichlet L-functions, or indeed any other L-function with negative coefficients provided they appear as a factor in a Dedekind zeta function.

2 Background on automorphic L-functions

In this section we collect some basic facts about the class of L-functions used in our t-aspect results. These can be found in many places, see for example [39, 41].

Let $L(s,\pi )$ be the L-function attached to an irreducible cuspidal automorphic representation $\pi$ of $\textrm{GL}(d)$ over $\mathbb {Q}$ normalised such that $\pi$ has unitary central character. In the region $\sigma >1$ we have

$$\begin{aligned} L(s,\pi ) = \sum _{n=1}^\infty \frac{A_\pi (n)}{n^s} = \prod _p\prod _{j=1}^d\bigg (1-\frac{\alpha _{\pi , j}(p)}{p^s}\bigg )^{-1} \end{aligned}$$

(2.1)

for some complex coefficients $A_\pi (n)$ and $\alpha _j(p)$. If $d=1$ and $\pi$ is the trivial representation then $L(s,\pi )$ is given by the Riemann zeta function. Otherwise, it extends to an entire function satisfying the functional equation

$$\begin{aligned} \Phi (s,\pi ):=N^{s/2}\gamma (s,\pi )L(s,\pi )=\epsilon _\pi {\overline{\Phi }}(1-s,\pi ) \end{aligned}$$

where $N\in \mathbb {N}$, $|\epsilon _\pi |=1$, ${\overline{\Phi }}(s,\pi )=\overline{\Phi (\overline{s},\pi )}$ and $\gamma (s,\pi )=\pi ^{-ds/2}\prod _{j=1}^d \Gamma \Big (\frac{s+\mu _{\pi , j}}{2}\Big )$ for some complex numbers $\mu _{\pi , j}$ satisfying $\Re \mu _{\pi , j}>-1$.

Applying Stirling’s formula to $\gamma (s,\pi )$ along with the Phragmen–Lindelöf principle, we see that

$$\begin{aligned} L(\sigma +it,\pi )\ll (N|t|^d)^{(1-\sigma )/2+\epsilon } \end{aligned}$$

(2.2)

in the strip $-\delta \leqslant \sigma \leqslant 1+\delta$ for large |t| (see Exercise 3 of Chapter 5, or Appendix 5.A.2, in [22] for example).

In our conditional results we make key use of Euler products, so we collect some further practical bounds here. For $\sigma >1$ on differentiating the Euler product we see

$$\begin{aligned} -\frac{L^\prime }{L}(s,\pi ) = \sum _{p^\ell \geqslant 2}\frac{\log p\sum _{j=1}^d \alpha _{\pi , j}(p)^\ell }{p^{\ell s}} =: \sum _{n\geqslant 2}\frac{\Lambda (n) a_\pi (n)}{n^s} \end{aligned}$$

(2.3)

where $\Lambda (n)$ is the von-Mangoldt function.

The generalised Ramanujan conjecture asserts that $|\alpha _{\pi , j}(p)|=1$ for all but a finite number of primes and satisfies $|\alpha _{\pi ,j}(p)|\leqslant 1$ elsewhere, although this remains open in general. Rudnick–Sarnak [41] have shown that

$$\begin{aligned} |\alpha _{\pi , j}(p)|\leqslant p^{1/2-1/(d^2+1)} \end{aligned}$$

for all primes p. This bound implies that

$$\begin{aligned} |a_\pi (p^\ell )|=\left| \sum _{j=1}^d \alpha _{\pi , j}(p)^\ell \right| \leqslant d p^{\ell (1/2-1/(d^2+1))} \end{aligned}$$

(2.4)

and, by (2.1), that

$$\begin{aligned} |A_\pi (n)| \leqslant \tau _d(n)n^{1/2-1/(d^2+1)} \end{aligned}$$

(2.5)

where $\tau _d$ is the generalised divisor function. Another useful bound is that when the degree satsifies $d\leqslant 3$,

$$\begin{aligned} |a_j(p^\ell )|\ll 1+ |a_j(p)|^\ell ,\qquad d\leqslant 3 \end{aligned}$$

(2.6)

for fixed prime powers $\ell$. This is given in the proof of Proposition 2.4 of [41].

In many situations, the assumption of the generalised Ramanujan conjecture can be replaced by Hypothesis H introduced by Rudnick–Sarnak [41], which states that for fixed $\ell \geqslant 2$,

$$\begin{aligned} \sum _{p}\frac{| a_\pi (p^\ell )|^2(\log p)^2}{p^\ell }<\infty . \end{aligned}$$

(2.7)

Clearly, this follows from the the generalised Ramanujan conjecture. Unconditionally, this was shown to hold for $d\leqslant 3$ by Rudnick–Sarnak [41] using (2.6) and for $d=4$ by Kim [27]. As a replacement for the generalised Ramanujan conjecture, Hypothesis H has previously been used in mean value results [39]. In our case it could be used to weaken the assumptions of Proposition 4 below, but at the cost of considerable extra technicalities. We settle for using (2.6), which restricts our unconditional results to $d\leqslant 3$ rather than $d\leqslant 4$, since in the end we require stronger coefficient bounds to deduce our final results.

The following three results are the key average properties we require for $a_\pi (p)$, all of which hold unconditionally when $d\leqslant 2$. The first is the orthogonality conjecture of Selberg, as shown in full generality by Liu–Ye [36] building on previous works [34, 35, 37].

Theorem

(Selberg’s orthogonality conjectures) Let $\pi$, $\pi ^\prime$ be two irreducible unitary cuspidal automorphic representations of $\textrm{GL}(d)$, $\textrm{GL}(d^\prime )$ over $\mathbb {Q}$, respectively. If (2.7) holds, then

$$\begin{aligned} \sum _{p\leqslant x}\frac{a_\pi (p)\overline{a_{\pi ^\prime }(p)}}{p} = {\left\{ \begin{array}{ll} \log \log x+O(1) &{} \quad \text {if } \pi \cong \pi ^\prime ,\\ O(1) &{} \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

(2.8)

In particular, (2.8) holds if $\max (d,d')\leqslant 4$ or on assuming the generalised Ramanujan conjecture.

This essentially implies the following bounds for our resonator sums.

Proposition 1

Let $\pi$, $\pi ^\prime$ be two irreducible unitary cuspidal automorphic representations of $\textrm{GL}(d)$, $\textrm{GL}(d^\prime )$ over $\mathbb {Q}$, respectively. If (2.7) holds then for large x, y,

$$\begin{aligned} \sum _{x<p\leqslant y}\frac{a_\pi (p)\overline{a_{\pi ^\prime }(p)}}{p\log p} = {\left\{ \begin{array}{ll} \frac{1}{\log x}- \frac{1}{\log y}+O\Big (\frac{1}{(\log x)^2}\Big ) &{} \quad \text {if } \pi \cong \pi ^\prime ,\\ O\Big (\frac{1}{(\log x)^2}\Big ) &{} \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

(2.9)

In particular, this holds if $\max (d,d^\prime )\leqslant 4$ or on assuming the generalised Ramanujan conjecture.

Proof

The sum is given by

$$\begin{aligned} \sum _{x<p\leqslant y}\frac{a_\pi (p)\overline{a_{\pi ^\prime }(p)}}{p\log p} = \sum _{x<n\leqslant y} \frac{\Lambda (n)a_\pi (n)\overline{a_{\pi ^\prime }(n)}}{n(\log n)^2} - \sum _{x<p^\ell \leqslant y,\ell \geqslant 2}\frac{a_\pi (p^\ell )\overline{a_{\pi ^\prime }(p^\ell )}}{\ell ^2p^\ell \log p}. \end{aligned}$$

To estimate the second sum we note that the contribution from $\ell \geqslant d^2+1$ can be bounded using (2.5) by

$$\begin{aligned} \ll \frac{1}{(\log x)^2}\sum _{p}\sum _{\ell \geqslant d^2+1}\frac{\log p}{p^{2\ell /d^2+1}}\ll \frac{1}{(\log x)^2}. \end{aligned}$$

For $\ell \leqslant d^2+1$ we use Cauchy–Schwarz and (2.7) to obtain

$$\begin{aligned} \ll \frac{1}{(\log x)^2}\sum _{\begin{array}{c} x<p^\ell \leqslant y\\ 2\leqslant \ell \leqslant d^2+1 \end{array}}\frac{\log p|a_\pi (p^\ell )\overline{a_{\pi ^\prime }(p^\ell )|}}{p^{\ell }}\ll \frac{1}{(\log x)^2}. \end{aligned}$$

To compute the first sum we apply partial summation along with the bounds of [36]:

$$\begin{aligned} S(x):=\sum _{n\leqslant x}\frac{(\log n)\Lambda (n)a_\pi (n)\overline{a_{\pi ^\prime }(n)}}{n}= {\left\{ \begin{array}{ll} \tfrac{1}{2}(\log x)^2+O(\log x) &{} \quad \text {if } \pi \cong \pi ^\prime , \\ O(\log x) &{} \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

$\square$

We shall use one more type of bound, which can be used to avoid the assumption of the generalised Ramanujan conjecture in some cases.

Theorem

(Fourth moment bounds) Suppose $d\leqslant 2$ or that $d=3$ and $\pi$ is self-dual. Then

$$\begin{aligned} \sum _{p\leqslant x}\frac{|a_\pi (p)|^4}{p}\ll \log \log x. \end{aligned}$$

(2.10)

Proof

For $d=1$ the result is clear. For $d=2$ this follows from the fact that

$$\begin{aligned} a_\pi (p)^4=2+a_{\textrm{Sym}^2 \pi }(p)+a_{\textrm{Sym}^4 \pi }(p), \end{aligned}$$

(see the proof of Corollary 2.15 of [5] for example) along with the bounds

$$\begin{aligned} \sum _{p\leqslant x}a_{\textrm{Sym}^k \pi }(p)/p\ll \log \log x, \ k=2, 4. \end{aligned}$$

When $d=3$ and $\pi$ is self-dual it is known (see Section 3.2 of [26]) that $L(s,\pi \times \pi \times \pi \times \pi )$ has a pole of order 3 at $s=1$. The result in this case therefore follows by Tauberian theorems.

$\square$

3 Simultaneous large values in t-aspect: set-up and proofs of Theorems 1 and 3

In this section we give the set-up for proving simultaneous large values in the t-aspect, state the required moment bounds and then complete the proofs of Theorems 1 and 3. Let $\pi _i$, $1\leqslant i\leqslant m$ be irreducible unitary cuspidal automorphic representations of $\textrm{GL}(d_i)$ over $\mathbb {Q}$ such that $\pi _i\not \cong \pi _j$ for $1\le i\not =j\le m$, respectively, and let

$$\begin{aligned} L_i(s)=L(s,\pi _i)=\sum _{n=1}^\infty \frac{A_{\pi _i}(n)}{n^s} \end{aligned}$$

be the associated L-functions. For brevity we denote $a_i(p)=a_{\pi _i}(p)=A_{\pi _i}(p)$.

To pick out simultaneous values we recall that if there exists a t such that

$$\begin{aligned} \prod _{i=1}^m |L_i(1/2+it)|^2-V\sum _{1\leqslant i\leqslant m}\prod _{\begin{array}{c} j=1\\ j\ne i \end{array}}^m|L_j(1/2+it)|^2 >0, \end{aligned}$$

then we must have $|L_i(1/2+it)|^2>V$ for all $1\leqslant i\leqslant m$. Write

$$\begin{aligned} L(s) = \prod _{i=1}^m L_i(s) = \sum _{n\geqslant 1}a(n)n^{-s} \end{aligned}$$

so that

$$\begin{aligned} a(p)=\sum _{i=1}^m A_{\pi _i}(p)=\sum _{i=1}^ma_{i}(p). \end{aligned}$$

We choose our resonator to pick out large values of L(s). Let $X=T^\Delta$ for some $\Delta <1$ to be chosen and denote

$$\begin{aligned} \mathcal {L}=\sqrt{\frac{1}{m}\log X\log \log X}. \end{aligned}$$

For small $\epsilon >0$ let

$$\begin{aligned} \mathcal {P}=\bigg \{\mathcal {L}^2<p\leqslant \exp ((\log \mathcal {L})^2): |a_i(p)|\leqslant (\log p)^{1-\epsilon } \text { for all } 1\leqslant i\leqslant m\bigg \}. \end{aligned}$$

We then define r(n) to be the multiplicative function supported on squarefree numbers for which

$$\begin{aligned} r(p)= {\left\{ \begin{array}{ll} a(p)\frac{\mathcal {L}}{\sqrt{p}\log p}{}, &{}\quad \text {for } p\in \mathcal {P,}\\ 0, &{} \quad \text {otherwise. } \end{array}\right. } \end{aligned}$$

and let

$$\begin{aligned} R(t)=\sum _{n\leqslant X}r(n)n^{-it}. \end{aligned}$$

(3.1)

By construction of $\mathcal {P}$ we note the important bounds

$$\begin{aligned} r(p)a_i(p)/p^{1/2}, |r(p)|^2=o(1) \end{aligned}$$

(3.2)

for any $1\leqslant i\leqslant m$ and $p\in \mathcal {P}$. With these bounds the computations for the Euler products acquired from the resonance method can proceed in the usual simple way. This is the reason for the restriction $|a_i(p)|\leqslant (\log p)^{1-\epsilon }$ in $\mathcal {P}$; without it such computations are much more involved e.g. see [5, Lemma 7.19].

With this set-up and the above notation we have the following propositions.

Proposition 2

For large T and $X=T^\Delta$ with $\Delta <1$, we have

$$\begin{aligned} \frac{1}{T} \int _{T}^{2T}\left| L\left( \tfrac{1}{2}+it\right) \right| ^2|R(t)|^2dt \gg \prod _p\bigg (1+|r(p)|^2+2(1+o(1))\frac{{r(p)}\overline{a(p)}}{\sqrt{p}}\bigg ). \end{aligned}$$

Proposition 3

Let $L_i(s)$ be a primitive Dirichlet L-function or the L-function of a (holomorphic or Maaß) cuspidal newform. Then for large T and $X=T^\Delta$ with $\Delta$ sufficiently small,

$$\begin{aligned} \frac{1}{T} \int _{T}^{2T}\left| L_i\left( \tfrac{1}{2}+it\right) \right| ^2|R(t)|^2dt \ll \prod _p\bigg (1+|r(p)|^2+2(1+o(1))\frac{\Re (r(p)\overline{a_i(p)})}{\sqrt{p}}\bigg ). \end{aligned}$$

If $L_i$ is a Dirichlet L-function, one can take $\Delta <\frac{17}{33}$ and if $L_i$ is a GL(2) L-function, we can take $\Delta <\frac{1/2-\theta }{3+\theta }$ where $\theta$ is the bound towards the Ramanujan conjecture for GL(2) Maaß Forms.

Proposition 4

Let $\pi _i$ be irreducible unitary cuspidal automorphic representations of $GL(d_i)$ over ${\mathbb {Q}}$ such that $\pi _i\not \cong \pi _j$ for $1\le i\not =j\le m$. Assume GRH for each $L_j(s)=L(s, \pi _j), j=1, \dots , m$ and if $d_j\geqslant 4$ assume the generalised Ramanujan conjecture for $L_j(s)$. Then for large T and $X=T^\Delta$ with $\Delta <1/2$, we have

$$\begin{aligned}&\frac{1}{T}\int _{T}^{2T}\prod _{\begin{array}{c} 1\leqslant j\leqslant m\\ j\ne i \end{array}}|L_j(\tfrac{1}{2}+it)|^{2}|R(t)|^2dt\\&\quad \ll \exp \bigg (\sqrt{\frac{\log T}{\log _2 T\log _3 T}}\bigg )\prod _p\bigg (1+|r(p)|^2+2(1+o(1))\frac{\Re ( r(p)\overline{b_i(p)})}{\sqrt{p}}\bigg ) \end{aligned}$$

where

$$\begin{aligned} b_i(p)=\sum _{j\ne i}a_j(p)=a(p)-a_i(p). \end{aligned}$$

Remark 3

The assumption of the generalised Ramanujan conjecture for $d_j\geqslant 4$ arises from Lemma 9 below. The remainder of the proof of Proposition 4 only requires the pointwise bounds on the coefficients given in (2.4) and (2.6). The generalised Ramanujan conjecture could be replaced with Hypothesis H of Rudnick–Sarnak [41], inequality (2.7), but at the cost of a more technical proof.

With the above propositions in hand, we turn to the proofs of Theorems 1 and 3.

Proof of Theorems 1 and 3

Define V as

$$\begin{aligned} V=\frac{\int _T^{2T}\left| L\left( \tfrac{1}{2}+it\right) \right| ^2|R(t)|^2 dt}{\sum _{i=1}^m\int _T^{2T} \prod _{j\ne i}\left| L_j\left( \tfrac{1}{2}+it\right) \right| ^2|R(t)|^2dt}. \end{aligned}$$

Then there exists $t\in [T,2T]$ satisfying (1.6) so that

$$\begin{aligned} \left| L_j\left( \tfrac{1}{2}+it\right) \right| ^2>V \text { for all }1\leqslant j\leqslant m. \end{aligned}$$

It remains to give a lower bound for V. From Propositions 2–4 and (3.2), we have that

$$\begin{aligned} V&\gg \exp \Big (-c\sqrt{\frac{\log T}{\log _2T\log _3T}}\Big )\\&\quad \times \exp \bigg (2(1+o(1))\min _{1\leqslant i\leqslant m}\sum _{p\in \mathcal {P}}\frac{\mathcal {L}(|a_i(p)|^2+\Re \overline{a_i(p)}\sum _{j\ne i}a_j(p))}{p\log p}\bigg ). \end{aligned}$$

Now let us remove the restriction on the size of the $|a_i(p)|$. For $1\le i\not =j\leqslant m$,

$$\begin{aligned}{} & {} \mathcal {L} \sum _{\begin{array}{c} \mathcal {L}^2<p\leqslant \exp ((\log \mathcal {L})^2)\\ |a_k(p)|>(\log p)^{1-\epsilon } \text {for some } k \end{array}}\frac{a_i(p)\overline{a_j(p)}}{p\log p}\\{} & {} \quad \ll \frac{\mathcal {L}}{(\log \mathcal {L})^{2-\epsilon }} \sum _{\begin{array}{c} \mathcal {L}^2<p\leqslant \exp ((\log \mathcal {L})^2) \end{array}}\frac{|a_i(p){a_j(p)}a_k(p)|}{p} \end{aligned}$$

which by Hölder’s inequality and (2.10) or the generalised Ramanujan conjecture is

$$\begin{aligned} \ll \frac{\mathcal {L}}{(\log \mathcal {L})^{2-\epsilon }}\log \log \mathcal {L}=o\Big (\frac{{\mathcal {L}}}{\log {\mathcal {L}}}\Big ). \end{aligned}$$

Thus we can extend the sum over $p\in \mathcal {P}$ to all $\mathcal {L}^2<p\leqslant \exp ((\log \mathcal {L})^2)$ with an acceptable error. Applying Proposition 1 gives

$$\begin{aligned} V\gg \exp \bigg (2(1+o(1))\sqrt{\frac{\frac{1}{m}\log X}{\log \log X}}\bigg ). \end{aligned}$$

$\square$

Remark 4

We believe that our choice of resonator coefficients is reasonably sharp. However, proving that they are truly optimal as in [44, Theorem 2.1], seems a more intricate optimization problem. In the case of two degree one L-functions, the mean values can be computed asymptotically leading to an optimization problem roughly of the shape

$$\begin{aligned} \frac{\displaystyle \sum _{m_1n_1=m_2n_2,\,m_j\leqslant T,n_j\leqslant X}\frac{a(m_1)\overline{a(m_2)}}{\sqrt{m_1m_2}}r(n_1)\overline{r(n_2)}}{\displaystyle \sum _{i=1}^2\sum _{m_1n_1=m_2n_2,\,m_j\leqslant T,n_j\leqslant X}\frac{a_i(m_1)\overline{a_i(m_2)}}{\sqrt{m_1m_2}}r(n_1)\overline{r(n_2)}}, \end{aligned}$$

(although in reality the mixed second moment leads to a sum of six terms with polar contributions which must be dealt with—see [4, 19, 21]). With suitable additional multiplicativity and positivity assumptions on r(n), this is essentially the optimization problem which is resolved in [44, Theorem 2.1], leading to our choice of the resonator. However, we have not pursued further in finding the optimal choice for r(n) in full generality and we leave it to the interested reader.

4 Lower bounds in t-aspect: Proof of Proposition 2

Let

$$\begin{aligned} \mathcal {I}:=\frac{1}{T}\int _{T}^{2T}|L(\tfrac{1}{2}+it)|^2|R(t)|^2dt. \end{aligned}$$

We aim to prove the lower bound

$$\begin{aligned} \mathcal {I} \gg \prod _p\bigg (1+|r(p)|^2+2(1+o(1))\frac{\overline{r(p)}a(p)}{\sqrt{p}}\bigg ). \end{aligned}$$

Let $\Phi$ be a smooth function supported on [1, 2] satisfying $0\leqslant \Phi (x)\leqslant 1$ so that

$$\begin{aligned} \mathcal {I} \geqslant \frac{1}{T}\int _\mathbb {R}|L(\tfrac{1}{2}+it)|^2|R(t)|^2\Phi (t/T)dt. \end{aligned}$$

Thus, by Cauchy–Schwarz

$$\begin{aligned} \mathcal {I} \geqslant \bigg |\frac{1}{T}\int _\mathbb {R}L\left( \tfrac{1}{2}+it\right) |R(t)|^2 \Phi (t/T)dt\bigg |^2\Big (\frac{1}{T}\int _\mathbb {R}|R(t)|^2\Phi (t/T)dt\Big )^{-1}. \end{aligned}$$

(4.1)

As usual, since $X=T^\Delta$ with $\Delta <1$, we find

$$\begin{aligned} \frac{1}{T}\int _\mathbb {R}|R(t)|^2\Phi (t/T)dt \sim {\hat{\Phi }}(0)\sum _{n\leqslant X} |r(n)|^2 \leqslant {\hat{\Phi }}(0)\prod _p (1+|r(p)|^2) \end{aligned}$$

(4.2)

and so it remains to compute

$$\begin{aligned} \frac{1}{T}\int _{\mathbb {R}}L\left( \tfrac{1}{2}+it\right) |R(t)|^2\Phi (t/T)dt. \end{aligned}$$

This has essentially been done by Aistleitner–Pańkowski [1], and so we only present the main details. The only difference is that they worked with the Selberg class where the generalised Ramanujan conjecture is assumed, although this has little effect on the arguments.

Applying the Mellin inversion formula

$$\begin{aligned} e^{-x}=\frac{1}{2\pi i }\int _{(c)}\Gamma (s) x^{-s}ds,\qquad c>0 \end{aligned}$$

and shifting contours to the left along with the convexity bounds (2.2), we find

$$\begin{aligned} L\left( \tfrac{1}{2}+it\right) =\sum _{n=1}^\infty \frac{a(n)}{n^{1/2+it}}e^{-n/Y}+O(1) \end{aligned}$$

where $Y=T^{d_L+\epsilon }$ and $d_L=\sum _{i=1}^m d_i$ is the degree of L(s). For $n>3Y\log Y$ we have $e^{-n/Y}\leqslant n^{-2}$ and hence

$$\begin{aligned} \sum _{n\geqslant 3Y\log Y}\frac{|a(n)|e^{-n/Y}}{n^{1/2}}\ll \sum _{n\geqslant 1} \frac{\tau _{d_L}(n)}{n^{2+1/(d^2+1)}} \ll 1 \end{aligned}$$

where we have used the coefficient bound for a(n) in (2.5). Thus we find

$$\begin{aligned} L(\tfrac{1}{2}+it)=\sum _{n\leqslant T^{d_L+2\epsilon }} \frac{a(n)}{n^{1/2+it}}e^{-n/Y}+O(1). \end{aligned}$$

From the rapid decay of ${\hat{\Phi }}$ and (4.2) we now have

$$\begin{aligned}{} & {} \frac{1}{T}\int _{\mathbb {R}}L\left( \tfrac{1}{2}+it\right) |R(t)|^2\Phi (t/T)dt \nonumber \\{} & {} \quad = {\hat{\Phi }}(0)\sum _{lm=n\leqslant X}\frac{a(l)r(m)\overline{r(n)}}{\sqrt{l}}e^{-l/Y} + O\big (\prod _p(1+|r(p)|^2)\big ). \end{aligned}$$

(4.3)

Since r(n) is supported on squarefree integers and $r(p)=a(p)\mathcal {L}/p^{1/2}\log p$ where it is non-zero, the summand of the main term is positive and hence we can bound it from below by

$$\begin{aligned} \frac{1}{2}{\hat{\Phi }}(0)\sum _{lm\leqslant X}\frac{a(l)r(m)\overline{r(lm)}}{\sqrt{l}} \end{aligned}$$

(4.4)

since $e^{-l/Y}\geqslant 1/2$ for $l\leqslant X$. Next we extend the sum $lm\le X$ to all l, m. By Rankin’s trick

$$\begin{aligned} \begin{aligned}&\frac{\sum _{lm>X}{a(l)r(m)\overline{r(lm)}}/{\sqrt{l}}}{\sum _{l,m}{a(l)r(m)\overline{r(lm)}}/{\sqrt{l}}} \ll X^{-\alpha }\prod _p \frac{1+|r(p)|^2p^{\alpha }+|r(p)a(p)|p^{-1/2 +\alpha }}{|1+|r(p)|^2+a(p)\overline{r(p)}p^{-1/2}|}\\&\quad \ll \exp \left( -\alpha \log X+\sum _{L^2<p\leqslant \exp ((\log L)^2)}|a(p)|^2\Big (\frac{\mathcal {L}^2}{p\log ^2 p}+\frac{\mathcal {L}}{p\log p}\Big )(p^{\alpha }-1)\right) \end{aligned}\nonumber \\ \end{aligned}$$

(4.5)

for any $\alpha >0$. Applying this along with the bound (2.9) and choosing $\alpha =1/(\log \mathcal {L})^3$, we find that (4.5) can be bounded by

$$\begin{aligned}{} & {} \ll \exp \Big (-\alpha \frac{\log X\log _3X}{\log _2X}+O(\alpha \frac{\mathcal L}{(\log \mathcal L)^2}+\alpha ^2 \mathcal L^2 \log \log \mathcal L)\Big )\nonumber \\{} & {} \quad \ll \exp \Big (- \frac{\log X}{(\log _2 X)^3}\Big ). \end{aligned}$$

(4.6)

Combining (4.3), (4.4), (4.5) and (4.6), we find that

$$\begin{aligned} \frac{1}{T}\int _{\mathbb {R}}L(\tfrac{1}{2}+it)|R(t)|^2\Phi (t/T)dt \gg \prod _p \bigg (1+|r(p)|^2+\frac{a(p)\overline{r(p)}}{\sqrt{p}}\bigg ). \end{aligned}$$

Applying this in (4.1) together with (3.2) we find that,

$$\begin{aligned} \mathcal {I}\gg \prod _p\frac{\Big (1+|r(p)|^2+\frac{a(p) \overline{r(p)}}{p^{1/2}}\Big )^2}{1+|r(p)|^2} \gg \prod _p \bigg (1+|r(p)|^2+2(1+o(1))\frac{a(p)\overline{r(p)}}{\sqrt{p}}\bigg ) \end{aligned}$$

as desired.

5 Unconditional upper bounds in t-aspect: Proof of Proposition 3

In this section we give the proof of Proposition 3 depending on whether $L_i$ is a Dirichlet L-function or a GL(2) L-function. For this we utilise the requisite twisted moment formulas which are known in these cases.

5.1 Dirichlet L-functions

In the case when $L_i(s)=L(s,\chi )$ where $\chi$ is a primitive Dirichlet character modulo q we have the following.

Lemma 5

Let $\chi$ be a primitive Dirichlet character modulo q. Let $R(t)=\sum _{n\le X}r(n)n^{-it}$ be as in (3.1). Let $\alpha , \beta$ be complex numbers such that $\alpha , \beta \ll 1/T$. Then for $X=T^\Delta$ with $\Delta <\frac{17}{33}$ there exists $\epsilon _\Delta >0$ such that

$$\begin{aligned}&\int _{T}^{2T}L\left( \tfrac{1}{2}+\alpha +it, \chi \right) L\left( \tfrac{1}{2}+ \beta -it, {\bar{\chi }}\right) |R(t)|^2dt+O(T^{1-\epsilon _\Delta })\\&\quad = \sum _{(hk,q)=1}\frac{(h,k)^{1+\alpha +\beta }r(h)\overline{\chi (h)}r(k) \chi (k)}{h^{1/2+\beta }k^{1/2+\alpha }}\\&\qquad \times \Big (L(1+\alpha +\beta , \chi _0)+\Big ( \frac{qt(h,k)^2}{2\pi hk}\Big )^{-\alpha -\beta } L(1-\alpha -\beta ,\chi _0)\Big ), \end{aligned}$$

where $\chi _0$ is the principal Dirichlet character modulo q.

Proof

The proof is similar to that [47, Theorem 1.1], but certain modifications are needed. First note that the condition $a(h)\ll h^\epsilon$ in the assumption [47, Theorem 1.1] does not hold in our case, however, we can modify the proof so that the conclusion still holds. More specifically, we still have [47, Eq. (5.28)] since $\sum _{u\sim U}\frac{\sqrt{u}r(u)}{u}\ll U\prod _{p}(1+r(p)\sqrt{p})\ll U \exp \Big (O\Big (\frac{{\mathcal {L}}}{\log {\mathcal {L}}^2}\Big )\Big )\ll U T^\epsilon$ and this causes an extra factor of $T^\epsilon$ in [47, Eq. (5.29)] which is acceptable. In the estimate [47, Eq. (5.37)], we use $\sum _{u\sim U}|\frac{\sqrt{u}r(u)}{u}|^2\ll \prod _{p}\Big (1+\frac{r(p)^2}{p}\Big )\ll \exp \Big (O\Big (\frac{{\mathcal {L}}^2}{(\log {\mathcal {L}}^2)^2}\Big )\Big )\ll T^\epsilon$, which is again acceptable and leads to [47, Eq. (5.38)]. The main term can be derived the same way in the proof of [47, Theorem 1.1] using [47, Proposition 3.1] (after correcting the typo in the exponent of (h, k)) or [10, Lemma 1 and Sect. 5]. $\square$

Let $X=T^{\Delta }$ with $\Delta < \frac{17}{33}$. We write

$$\begin{aligned}&\int _{T}^{2T}\left| L\left( \tfrac{1}{2}+it,\chi \right) \right| ^2|R(t)|^2dt \\&\quad = \lim _{\alpha ,\beta \rightarrow 0} \int _T^{2T}L\left( \tfrac{1}{2}+\alpha +it,\chi \right) L \left( \tfrac{1}{2}+\beta -it,\overline{\chi }\right) |R(t)|^2dt. \end{aligned}$$

By a residue calculation, we see that

$$\begin{aligned}&{L(1+\alpha +\beta ,\chi _0)}+\bigg (\frac{qt}{2\pi HK}\bigg )^{-\alpha -\beta }{L(1-\alpha -\beta ,\chi _0)}\\&\quad = -\frac{\left( \frac{qt}{2\pi HK}\right) ^{-(\alpha +\beta )/2}}{(2\pi i)^2}\int _{|z_j|=\tfrac{2^j}{\log T}} L(1+z_1-z_2,\chi _{0})(z_1-z_2)^2 \bigg (\frac{qt}{2\pi HK}\bigg )^{\tfrac{z_1-z_2}{2}}\\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \times \prod _{j=1}^2\frac{dz_j}{(z_j-\alpha )(z_j+\beta )}. \end{aligned}$$

Applying this in Lemma 5 with $HK=hk/(h,k)^2$, we obtain

$$\begin{aligned}&\int _T^{2T}\left| L\left( \tfrac{1}{2}+it,\chi \right) \right| ^2|R(t)|^2dt+O(T^{1-\epsilon _{\Delta }})\\&\quad =\frac{1}{(2\pi i)^2}\int _{|z_j|=\tfrac{2^j}{\log T}}L(1+z_1+z_2,\chi _{0})G_X(z_1,z_2)(z_1+z_2)^2\nonumber \\&\qquad \qquad \qquad \quad \times \bigg (\int _T^{2T}\bigg (\frac{qt}{2\pi }\bigg )^{\tfrac{z_1 +z_2}{2}}dt\bigg ) \prod _{j=1}^2\frac{dz_j}{z_j^2},\nonumber \end{aligned}$$

(5.1)

where

$$\begin{aligned} G_X(z_1,z_2)=\sum _{h,k\leqslant X}\frac{\overline{\chi }(h/(h,k)){\chi } (k/(h,k))r(h)\overline{r(k)}}{(hk)^{1/2+(z_1+z_2)/2}}(h,k)^{1+z_1+z_2}. \end{aligned}$$

By Rankin’s trick we see that

$$\begin{aligned} G_X(\underline{z})=N_X(\underline{z})+O(\mathcal {E}_X(\underline{z})) \end{aligned}$$

(5.2)

where

$$\begin{aligned} N_X(\underline{z})&= \sum _{h,k}\frac{\overline{\chi }(h/(h,k)){\chi }(k/(h,k))r(h) \overline{r(k)}}{(hk)^{(1+z_1+z_2)/2}}(h,k)^{1+z_1+z_2} \\&= \prod _p \bigg (1+|r(p)|^2+2\frac{\Re r(p)\overline{\chi }(p)}{p^{(1+z_1+z_2)/2}}\bigg ) \end{aligned}$$

and

$$\begin{aligned} \mathcal {E}_X(\underline{z})=X^{-\alpha }\prod _p \bigg (1+|r(p)|^2p^\alpha +\frac{ |r(p)|}{p^{(1+\Re z_1+\Re z_2)/2}}(p^\alpha +1)\bigg ) \end{aligned}$$

for any $\alpha >0$. Using the approximation (5.2) in (5.1) and calculating the integral of the main term via residues at $z_j=0$, we find that the leading term in (5.1) is of size

$$\begin{aligned} T(\log T )N_X(\underline{0}) \end{aligned}$$

with the lower order terms involving partial derivatives of $N_X(\underline{z})$. To estimate these we note

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial z_1}N_X(\underline{z})\bigg |_{\underline{z}=\underline{0}}&\ll N_X(\underline{0})\sum _p\frac{|r(p)|\log p/p^{1/2}}{|1+|r(p)|^2+2\Re \frac{r(p){\overline{\chi (p)}}}{p^{1/2}}|}\\&\ll N_X(\underline{0})\sum _{\mathcal {L}^2<p\leqslant \exp ((\log \mathcal {L})^2)}\frac{\mathcal {L}|a(p)|}{p} \ll N_X(\underline{0})(\log X)^{1/2+\epsilon } \end{aligned} \end{aligned}$$

(5.3)

by Cauchy–Schwarz, (3.2) and (2.8). Note that the integrand of (5.1) is $\ll (\log T)^3$ and so trivial estimation of the contribution from $\mathcal {E}_X(\underline{z})$ to this integral gives

$$\begin{aligned} \mathcal {J}_i\leqslant C(\log T) N_X(\underline{0}) +O((\log T)\mathcal {E}(X))+O(T^{-\epsilon _\Delta }), \end{aligned}$$

where $\mathcal {E}(X)=\mathcal {E}_X(-2/\log T,-4/\log T)$. Now

$$\begin{aligned} \begin{aligned} \frac{\mathcal {E}(X)}{N_X(\underline{0})}&\ll \exp \left( -a\log X + \sum _{p} |r(p)|^2(p^\alpha -1)+O\left( \sum _p|r(p)|p^{-1/2}\right) \right) \\&\leqslant \exp \left( -a\log X + \sum _{\mathcal {L}^2<p\leqslant \exp ((\log \mathcal {L})^2)} (p^\alpha -1)\frac{\mathcal {L}^2|a(p)|^2}{p\log ^2 p}+O\left( \frac{\mathcal {L}}{\log \mathcal {L}}\right) \right) \end{aligned} \end{aligned}$$

which, similarly to (4.5), is o(1) on choosing $\alpha =1/(\log \mathcal {L})^3$. Thus, we find that when $L_i(s)=L(s,\chi )$,

$$\begin{aligned} \mathcal {J}_i\ll (\log T) N_X(\underline{0}) \ll \log T\prod _p \bigg (1+|r(p)|^2+2\Re \frac{ r(p)\overline{a_i}(p)}{p^{1/2}}\bigg ), \end{aligned}$$

which completes the proof of Proposition 3 for the case for Dirichlet L-functions after noting that the factor of $\log T$ can be absorbed into the o(1) term in the product.

5.2 $GL(2)\,L$-functions

Here we are in the case where $L_i=L(s,f)$ is the L-function of a primitive cusp form f. Let $\Phi : {\mathbb {R}}\rightarrow {\mathbb {R}}$ be a smooth function supported on [1/4, 2] satisfying $\Phi (x)\geqslant 1$ on $x\in [1,2]$ along with the bounds $\Phi ^{(j)}(x)\ll (\log T)^{j}$ for each $j\ge 0$.

Lemma 6

Let $L(s,f)=\sum _{n\geqslant 1}\lambda _f(n)n^{-s}$ be the L-function of a Hecke newform (holomorphic or Maaß) of level N. Let $\alpha , \beta$ be complex numbers satisfying $\alpha , \beta \ll 1/\log T$. Let $(h,k)=1$ and $\Phi$ be as above. Then we have

$$\begin{aligned}&\int _\mathbb {R}L\left( \tfrac{1}{2}+\alpha +it,f\right) L\left( \tfrac{1}{2}+\beta -it,{f}\right) (h/k)^{-it}\Phi \big (t/T\big )dt\\&\quad = \frac{1}{{h^{1/2+\beta }k^{1/2+\alpha }}}\int _\mathbb {R} \Big (L^*(1+\alpha +\beta ,f\otimes f)Z_{\alpha ,\beta }(h,k)\\&\qquad + \bigg (\frac{t\sqrt{N}}{2\pi \sqrt{hk}}\bigg )^{-2(\alpha +\beta )}L^*(1-\alpha -\beta ,f\otimes f)Z_{-\beta ,-\alpha }(h,k)\Big ) \Phi \left( t/T\right) dt\\&\qquad +O\left( (hk)^{1/2+\epsilon }T^{1/2+\theta +\epsilon }\right) \end{aligned}$$

where $L^*(s, f\otimes f)=\sum _{n}\lambda _f(n)^2 n^{-s}$ and

$$\begin{aligned} Z_{\alpha , \beta }(h,k)=\prod _{p\mid hk}(1-p^{-2(1+\alpha +\beta )})^{-1}\Big (\lambda _f(p)- \frac{\lambda _f(p)}{p^{1+\alpha +\beta }}\Big ). \end{aligned}$$

(5.4)

Proof

This follows from [2], which improves earlier results for the holomorphic case in [3, 31]. Using [2, Proposition 3.4], we have for $|\alpha + \beta |\gg 1/\log T$

$$\begin{aligned}&\int _\mathbb {R}L\left( \tfrac{1}{2}+\alpha +it,f\right) L\left( \tfrac{1}{2}+\beta -it,{f}\right) (h/k)^{-it}\Phi (t/T)dt +O((hk)^{1/2}T^{1/2+\theta +\epsilon })\\&\quad =\sum _{hm=kn}\frac{\lambda _f(m)\lambda _f(n)}{m^{1/2 +\alpha }n^{1/2+\beta }}\int _\mathbb {R}V_{\alpha ,\beta }(mn,t)\Phi (t/T)dt\\&\qquad + \sum _{hm=kn}\frac{\lambda _f(m)\lambda _f(n)}{m^{1/2 -\beta }n^{1/2-\alpha }}\int _\mathbb {R}X_{\alpha ,\beta ,t}V_{-\beta ,-\alpha }(mn,t)\Phi (t/T)dt \end{aligned}$$

where

$$\begin{aligned} V_{\alpha ,\beta }(x) = \frac{1}{2\pi i }\int _{1-i\infty }^{1+i\infty } \frac{G(s)}{s}x^{-s} g_{\alpha ,\beta }(s,t)ds \end{aligned}$$

with

$$\begin{aligned} G(s)= e^{s^2}\frac{(\alpha +\beta )^2-(2s)^2}{(\alpha +\beta )^2}, \end{aligned}$$

and where $g_{\alpha ,\beta }(s,t)$ and $X_{\alpha ,\beta ,t}$ are ratios of gamma factors satisfying

$$\begin{aligned}{} & {} g_{\alpha ,\beta }(s,t) = \bigg (\frac{t\sqrt{N}}{2\pi }\bigg )^{2s} \Big (1+O\big (\frac{|s|^2}{t}\big )\Big ),\nonumber \\{} & {} \quad \quad X_{\alpha ,\beta ,t} = \bigg (\frac{t\sqrt{N}}{2\pi }\bigg )^{-2(\alpha +\beta )}\Big (1+O\big (\frac{|\alpha ^2-\beta ^2|}{t}\big )\Big ) \end{aligned}$$

(5.5)

(see e.g. Lemma 2 of [3]). Using the definition of $V_{\alpha ,\beta }(x)$ and moving the m, n-sum inside, we encounter the Dirichlet series

$$\begin{aligned} \sum _{hm=kn}\frac{\lambda _f(m)\lambda _f(n)}{m^{1/2+\alpha +s}n^{1/2+\beta +s}}&= \frac{1}{k^{1/2+\alpha +s}h^{1/2+\beta +s}}\sum _{l\geqslant 1}\frac{\lambda _f(kl)\lambda _f(hl)}{l^{1+\alpha +\beta +2s}} \end{aligned}$$

since $(h,k)=1$. Using multiplicativity and Hecke relations (see e.g. [5, Proof of Lemma 7.9]), we see that

$$\begin{aligned} D(s;h,k)&:=\sum _{l\ge 1}\frac{\lambda _f(kl)\lambda _f(hk)}{l^s}\\&=L^*(s,f\otimes f)\prod _{p\mid hk}(1-p^{-2s})^{-1}\Big (\lambda _f(p)-\frac{\lambda _f(p)}{p^s}\Big ) \end{aligned}$$

where $L^*(s,f\otimes f)=\sum _{n\ge 1}\lambda _f(n)^2n^{-s}$. Shifting the contour to $\Re (s)=-1/4+\epsilon$ we encounter a simple pole at $s=0$ which gives the main term. The contribution from the remaining contour is seen to be $\ll T^{1/2}(hk)^{-1/4+\theta +\epsilon }$ by (5.5), the rapid decay of G(s), the convexity bound $L^*(1/2+\epsilon +iy,f\otimes f)\ll (1+|y|)^{1+\epsilon }$, and the bound

$$\begin{aligned} \prod _{p\mid hk}\Big (\lambda _f(p)+O\left( \frac{\lambda _f(p)}{p^{1/2 +2\epsilon }}\right) \Big ) \ll (hk)^{\theta +\epsilon }. \end{aligned}$$

By analytic continuation, the result hold for $\alpha , \beta \ll 1/\log T$. $\square$

To complete the proof of Proposition 3 for the case of GL(2) L-functions, we follow the same argument as before in the case for Dirichlet L-functions after replacing Lemma 5 by Lemma 6 and $G_X(z_1, z_2)$ by

$$\begin{aligned} H_X(z_1,z_2) = \sum _{h,k\leqslant X}\frac{r(h)\overline{r(k)}Z_{z_1,z_2}(h/(h,k),k/(h, k))}{(hk)^{(1+z_1+z_2)/2}}(h,k)^{1+z_1+z_2} \end{aligned}$$

where $Z_{z_1, z_2}$ is defined in (5.4). Note that we have

$$\begin{aligned} Z_{\underline{0}}(h,k)=\prod _{p\mid hk}(1-p^{-2})^{-1}\Big (\lambda _f(p)-\frac{\lambda _f(p)}{p}\Big ) \end{aligned}$$

and that $\lambda _f(p)(1-1/p)=a_i(p)(1+o(1))$ for large p. Thus, the main contribution to $\mathcal {J}_i$ in this case, aside from some factors of $\log T$ which can be absorbed into the o(1), is

$$\begin{aligned}{} & {} \sum _{h,k}\frac{r(h)\overline{r(k)}Z_{\underline{0}}(h/(h,k),k/(h,k))}{(hk)^{1/2}}(h,k)\\{} & {} \quad =\prod _p\bigg (1+|r(p)|^2+2(1+o(1))\Re \frac{r(p)\overline{a_i(p)}}{p^{1/2}}\bigg ) \end{aligned}$$

as required. Again, the lower order terms coming from partial derivatives can be bounded similarly to (5.3) using Proposition 1 whilst the error from the tail sums $h>X$, $k>X$ are also of a lower order by similar arguments to before (again using Proposition 1).

6 Conditional upper bounds in t-aspect: Proof of Proposition 4

6.1 Upper bounds for the logarithm of the product of L-functions

Let $\pi _i$ be irreducible unitary cuspidal automorphic representations of $GL(d_i)$ over ${\mathbb {Q}}$ such that $\pi _i\not \cong \pi _j$ for $1\le i\not =j\le m$. For a given $1\leqslant i\leqslant m$ write

$$\begin{aligned} M(s)=M_i(s)=\prod _{\begin{array}{c} j=1\\ j\ne i \end{array}}^m L_j(s)=\sum _{n=1}^\infty \frac{B_i(n)}{n^s} \end{aligned}$$

so that

$$\begin{aligned} \frac{M'(s)}{M(s)}=\sum _{n}\frac{\Lambda (n)b(n)}{n^s} \end{aligned}$$

where

$$\begin{aligned} b(n)=b_i(n)=\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^ma_{\pi _j}(n) \end{aligned}$$

where $a_{\pi }(n)$ are defined as in (2.3). The goal is to show that

$$\begin{aligned}&\frac{1}{T}\int _{T}^{2T}|M(\tfrac{1}{2}+it)|^{2}|R(t)|^2dt \\&\quad \ll \exp \bigg (\sqrt{\frac{\log T}{\log _2 T\log _3 T}}\bigg )\prod _p\bigg (1+|r(p)|^2+2\Re \frac{r(p)\overline{b(p)}}{p^{1/2}}\bigg ). \end{aligned}$$

We plan to compute the integral over t by using Harper’s method [17]. A key estimate will be

$$\begin{aligned} \sum _{p\leqslant x}\frac{|b(p)|^2}{p}=(m-1)\log \log x+O(1) \end{aligned}$$

(6.1)

which follows by (2.8) and the assumption on $\pi _j$. To measure the size of exceptional sets where Dirichlet polynomials obtain large values we need the following standard lemma.

Lemma 7

[45, Lemma 3] Let T be large and let $2\leqslant x\leqslant T$. Let k be a natural number such that $x^{k}\leqslant T$. Then for any complex numbers c(p) we have

$$\begin{aligned} \frac{1}{T}\int _{T}^{2T}\bigg |\sum _{p\leqslant x}\frac{c(p)}{p^{1/2+it}}\bigg |^{2k}dt\ll k!\bigg (\sum _{p\leqslant x}\frac{|c(p)|^2}{p}\bigg )^k. \end{aligned}$$

To obtain an upper bound for $|M|^2$ we apply the following generalisation of Soundararajan’s result for the Riemann zeta function [45, Proposition] due to Chandee [9].

Lemma 8

[9, Theorem 2.1] Assume GRH holds for $L(s,\pi _j)$, $1\leqslant j\leqslant m$, $j\ne i$. Then for $2\leqslant Z\leqslant T^2$ and $t\in [T,2T]$ we have

$$\begin{aligned} \log |M(1/2+it)|\leqslant \Re \sum _{n\leqslant Z}\frac{\Lambda (n)b(n)w_Z(n)}{n^{1/2+it}\log n}+C_M\frac{\log T}{\log Z}+O(1) \end{aligned}$$

for some positive constant $C_M$ where

$$\begin{aligned} w_Z(n)=n^{-1/2\log Z}\Big (1-\frac{\log n}{\log Z}\Big )\leqslant 1. \end{aligned}$$

Since we are interested in extreme values, powers of $\log T$ will not meaningfully affect our final bound and so we first trivially bound the sum over prime powers.

Lemma 9

Assume GRH holds for $L(s,\pi _j)$, $1\leqslant j\leqslant m$, $j\ne i$ and that the generalised Ramanujan conjecture holds for $L(s,\pi _j)$ if $d_j\geqslant 4$. Then for $2\leqslant Z\leqslant T^2$ and $t\in [T,2T]$ we have

$$\begin{aligned} \log |M(1/2+it)|\leqslant \Re \sum _{\begin{array}{c} p \leqslant Z \end{array}}\frac{b(p)w_Z(p)}{p^{1/2+it}}+C_M\frac{\log T}{\log Z}+O(\log \log Z). \end{aligned}$$

(6.2)

Proof

The terms in Lemma 8 with powers $\ell >d^2+1$ with $d=\max _j d_j$ are

$$\begin{aligned} \ll \sum _{\begin{array}{c} p^\ell \leqslant Z\\ \ell > d^2+1 \end{array}}\frac{|b(p^\ell )|}{p^{\ell /2}}\ll \sum _{p}\frac{1}{p^{\ell /(d^2+1)}}\ll 1 \end{aligned}$$

which follows from (2.4) and the fact that $b(p^\ell )=\sum _{j\ne i}a_{j}(p^\ell )$. For $2\leqslant \ell \leqslant d^2+1$ we note that if $d_j\leqslant 3$ then (2.6) gives

$$\begin{aligned} |a_j(p^\ell )|\ll 1+ |a_j(p)|^\ell . \end{aligned}$$

If $d_j\geqslant 4$ then the generalised Ramanujan conjecture implies $|a_j(p^\ell )|\leqslant d_j$. Thus the prime squares contribute

$$\begin{aligned} \ll \sum _{p\leqslant Z^{1/2}}\frac{|b(p^2)|}{p} \ll \log \log Z \end{aligned}$$

by (2.8) whilst by (2.4) we have

$$\begin{aligned} \sum _{p}\frac{|b(p)|^\ell }{p^{\ell /2}}\ll \sum _{j\ne i}\sum _{p}\frac{|a_j(p)|^2}{p^{1+(\ell -2)/(d_j^2+1)}}. \end{aligned}$$

Since the Rankin–Selberg L-function is convergent for $\sigma >1$, this last sum is bounded for $\ell \geqslant 3$. $\square$

6.2 Initial splitting and the exceptional set

Following [17], the aim is to take a reasonably large Z in (6.2) and split the sum over primes into pieces with small variance so that for typical t their exponential can be approximated by a short truncated Taylor series. For us, the choice at where to begin this splitting is dictated by the support of the resonator coefficients, namely $p\leqslant \exp ((\log \mathcal {L})^2)$—the main interaction between M and the resonator will come from this piece. This gives a large chunk of primes in the first sum but as the following lemma shows, this is just about affordable and the exceptional set of large values of this sum is sufficiently small in measure.

Lemma 10

For $Z\leqslant X$ let

$$\begin{aligned} E=\Big \{t\in [T,2T]: \Big |\sum _{\begin{array}{c} p\leqslant \exp ((\log \mathcal {L})^2) \end{array}}\frac{b(p)w_Z(p)}{ p^{1/2+it}}\Big |\geqslant \frac{\log T}{100(\log \mathcal {L})^2}\Big \}. \end{aligned}$$

Then

$$\begin{aligned} \mu (E)\ll T\exp \Big (-4(1-o(1))\frac{\log T}{\log _2 T}\Big ). \end{aligned}$$

In particular, under the assumptions in Proposition 4 for $X= T^{\Delta }$ with $\Delta <1/2$, we have

$$\begin{aligned} \int _{E}|M(\tfrac{1}{2}+it)|^2|R(t)|^2dt=o(T). \end{aligned}$$

Proof

By Lemma 7 along with (6.1) we have

$$\begin{aligned} \frac{1}{T}\mu ({E})&\ll k!\bigg (\sum _{\begin{array}{c} p\leqslant \exp ((\log \mathcal {L})^2) \end{array}}\frac{|b(p)|^2}{ p}\bigg )^k \bigg (\frac{\log T}{100(\log \mathcal {L})^2}\bigg )^{-2k}\\&\ll k^{1/2}\bigg (\frac{Ck\log _3 T}{(\log T/(\log \mathcal {L})^2)^2}\bigg )^k \end{aligned}$$

for some C provided $k\leqslant \frac{\log T}{(\log \mathcal {L})^2}$. Choosing $k=\frac{\log T}{(\log \mathcal {L})^2}=4(1+o(1))\frac{\log T}{(\log _2 T)^2}$ this is

$$\begin{aligned} \ll \bigg (\frac{C^\prime (\log _2 T)^2\log _3 T}{\log T}\bigg )^{4(1+o(1))\log T/(\log _2 T)^2} \leqslant \exp \left( -4(1-o(1))\frac{\log T}{\log _2T}\right) \end{aligned}$$

giving the first part of the lemma.

By Hölder’s inequality we have

$$\begin{aligned}&\int _{{E}} |M(\tfrac{1}{2}+it)|^2 |R(t)|^2dt\\&\quad \leqslant \mu (E)^{1/4} \bigg (\int _{T}^{2T}|M(\tfrac{1}{2}+it)|^{8}dt\bigg )^{1/4} \bigg (\int _{T}^{2T}|R(t)|^4dt\bigg )^{1/2}\\&\quad \ll T^{1/2}\exp \bigg (-(1+o(1))\frac{\log T}{\log _2 T}\bigg )\bigg (\int _{T}^{2T}|R(t)|^4dt\bigg )^{1/2} \end{aligned}$$

on applying the conditional bound $\int _{T}^{2T} |M(1/2+it)|^8dt\ll T(\log T)^{O(1)}$ which follows from [39].

By the mean value theorems for Dirichlet polynomials, for $X\leqslant T^{1/2-\epsilon }$ we have

$$\begin{aligned} \int _{T}^{2T}|R(t)|^4dt&\ll T\sum _{\begin{array}{c} n_1n_2=n_3n_4\\ n_j\leqslant X \end{array}}|r(n_1)r(n_2)r(n_3)r(n_4)|\\&\leqslant T\prod _p \big (1+4|r(p)|^2+|r(p)|^4\big )\\&\leqslant T\exp \bigg (4\sum _{p}|r(p)|^2 \bigg ) \ll T\exp \bigg (4\sum _{\mathcal {L}^2<p}\frac{\mathcal {L}^2|a(p)|^2}{p\log ^2 p} \bigg )\\&\ll T\exp \bigg (4(1+o(1))\frac{\log X}{\log _2 X}\bigg )\ll T\exp \Big ((2-\epsilon +o(1))\frac{\log T}{\log _2T}\Big ) \end{aligned}$$

by (2.9). $\square$

6.3 Remaining splittings and an inequality for $|M|^2$

A key point is that on the set $[T,2T]\backslash E$ the exponential of the sum in the above lemma can be approximated by a short truncated Taylor series to give a Dirichlet polynomial of length $\leqslant T^{1/10}$. Our choice of parameters throughout will be dictated by the need to have short Dirichlet polynomials whilst also having small exceptional sets.

For integer $\mathfrak {i}\geqslant 0$ let

$$\begin{aligned} Z_\mathfrak {i}=\exp (e^\mathfrak {i}(\log \mathcal {L})^2),\qquad \qquad Z_{-1}=1. \end{aligned}$$

Let J be the minimal integer such that $Z_J\geqslant \exp ({2}{C_M}\sqrt{\log T\log _2 T\log _3 T})$, so that $J=(\tfrac{1}{2}+o(1))\log \log T$ and note

$$\begin{aligned} C_M\frac{\log T}{\log Z_J} \leqslant \frac{1}{2}\sqrt{\log T/\log _2 T\log _3 T}. \end{aligned}$$

By a slight abuse of notation we write $w_{Z_\mathfrak {j}}(p)$ as $w_\mathfrak {j}(p)$. Let

$$\begin{aligned} P_{\mathfrak {i},\mathfrak {j}}(t)=\sum _{Z_{\mathfrak {i}-1}<p\leqslant Z_\mathfrak {i}}\frac{b(p)w_\mathfrak {j}(p)}{p^{1/2+it}}. \end{aligned}$$

so that

$$\begin{aligned} \sum _{p\leqslant Z_\mathfrak {j}}\frac{b(p)w_\mathfrak {j}(p)}{p^{1/2+it}}=\sum _{i=0}^\mathfrak {j} P_{\mathfrak {i},\mathfrak {j}}(t). \end{aligned}$$

Set

$$\begin{aligned} \ell _\mathfrak {i}=\frac{\log T}{100(\log \mathcal {L})^2}e^{-5\mathfrak {i}/4},\qquad r_\mathfrak {i}=\frac{\log T}{e^\mathfrak {i}(\log \mathcal {L})^{2+\epsilon }}. \end{aligned}$$

We remark that $P_{\mathfrak {i},\mathfrak {j}}(t)^{10\ell _\mathfrak {i}}$ is a Dirichlet polynomial of length $Z_{\mathfrak {i}}^{10\ell _{\mathfrak {i}}}=T^{e^{-\mathfrak {i}/4}/10}$. By Lemma 7 and (6.1) the measure of the set where $|P_{\mathfrak {i},\mathfrak {j}}(t)|\geqslant \ell _\mathfrak {i}$ is, for any $k\leqslant \log T/(e^\mathfrak {i}(\log \mathcal {L})^2)$,

$$\begin{aligned} \ll k^{1/2}\bigg (\frac{k\sum _{Z_{\mathfrak {i}-1}<p\leqslant Z_\mathfrak {i}}|b(p)|^2/p}{e\ell _\mathfrak {i}^2}\bigg )^k \ll \bigg (\frac{ck}{\ell _\mathfrak {i}^2}\bigg )^k \ll \exp \Big (-\tfrac{c^\prime \log T}{e^\mathfrak {i}(\log _2 T)^{1+\epsilon }}\Big )\nonumber \\ \end{aligned}$$

(6.3)

for some constants $c,c^\prime$ on choosing $k=r_\mathfrak {i}$ (we don’t choose k as large as possible because we will need shorter Dirichlet polynomials later and this bound is sufficient). Note this bound kills

$$\begin{aligned} \exp \Big (C_M\frac{\log T}{\log Z_{\mathfrak {i}-1}}\Big )=\exp \Big ((1+o(1))\frac{4C_M\log T}{e^{\mathfrak {i}-1}(\log _2 T)^2}\Big ) \end{aligned}$$

(6.4)

which will be the extra term acquired from applying the inequality (6.2) at the $\mathfrak {i}$th step. As a final remark on our parameter choices: the reason for factor $e^{-5\mathfrak {i}/4}$ in $\ell _{\mathfrak {i}}$ is, first of all, so that we have a polynomial of length $T^{e^{-\mathfrak {i}/4}/10}$ which, after taking the product over all $\mathfrak {i}$, is still short (see (6.8) below). A factor of $e^{-c\mathfrak {i}}$, $c>1$, is required to have the decay in the exponent of T however if $c>3/2$ then $\ell _J$ would not be large enough to guarantee (6.3). The reason for the factor of $e^{-\mathfrak {i}}$ in $r_\mathfrak {i}$ is so that (6.3) is comparable with (6.4) for all $\mathfrak {i}$.

Now, by Stirling’s formula for $|z|\leqslant L$ we have

$$\begin{aligned} e^z=(1+O(e^{-9L}))\sum _{m\leqslant 10L}\frac{z^m}{m!}. \end{aligned}$$

Therefore, if $|P_{\mathfrak {i},\mathfrak {j}}(t)|\leqslant \ell _i$ we have

$$\begin{aligned} \exp (P_{\mathfrak {i},\mathfrak {j}}(t))=(1+O(e^{-9\ell _\mathfrak {i}}))\sum _{m\leqslant 10\ell _\mathfrak {i}}\frac{P_{\mathfrak {i},\mathfrak {j}}(t)^m}{m!}. \end{aligned}$$

The multinomial theorem gives

$$\begin{aligned} P_{\mathfrak {i},\mathfrak {j}}(t)^m=m!\sum _{\begin{array}{c} \Omega (n)=m\\ p|n\implies Z_{\mathfrak {i}-1}<p\leqslant Z_\mathfrak {i} \end{array}}\frac{c(n)W_\mathfrak {j}(n)\mathfrak {g}(n)}{n^{1/2+it}} \end{aligned}$$

where

$$\begin{aligned} c(n)=\prod _{p^{\alpha _p}||n}b(p)^{\alpha _p},\qquad W_\mathfrak {j}(n)=\prod _{p^{\alpha _p}||n}w_\mathfrak {j}(p)^{\alpha _p} \end{aligned}$$

(6.5)

are the completely multiplicative extensions of b(p) and $w_\mathfrak {j}(p)$ to the integers and $\mathfrak {g}$ is the multiplicative function for which

$$\begin{aligned} \mathfrak {g}(p^\alpha )=\frac{1}{\alpha !}. \end{aligned}$$

(6.6)

Thus, if we denote

$$\begin{aligned} {\mathcal {N}}_{\mathfrak {i},\mathfrak {j}}(t)=\sum _{\begin{array}{c} \Omega (n)\leqslant 10\ell _\mathfrak {i}\\ p|n\implies Z_{\mathfrak {i}-1}< p\leqslant Z_\mathfrak {i} \end{array}}\frac{c(n)W_\mathfrak {j}(n)\mathfrak {g}(n)}{n^{1/2+it}} \end{aligned}$$

then on such a set of t we have

$$\begin{aligned} \exp \big (2\Re P_{\mathfrak {i},\mathfrak {j}}(t)\big )=(1+O(e^{-9\ell _\mathfrak {i}}))|\mathcal {N}_{\mathfrak {i},\mathfrak {j}}(t)|^2. \end{aligned}$$

Accordingly, if t is such that $|{P}_{\mathfrak {i},\mathfrak {j}}(t)|\leqslant \ell _\mathfrak {i}$ for all $0\leqslant \mathfrak {i}\leqslant \mathfrak {j}$ then

$$\begin{aligned} \exp \Big (2\Re \sum _{p\leqslant Z_\mathfrak {j}}\frac{w_\mathfrak {j}(p)}{p^{{1}/{2}+it}}\Big ) =(1+o(1)) \prod _{\mathfrak {i}=0}^\mathfrak {j}\big |{\mathcal {N}}_{\mathfrak {i},\mathfrak {j}}(t)\big |^2 \end{aligned}$$

(6.7)

since $\sum _{\mathfrak {i}=0}^\mathfrak {j}e^{-9\ell _\mathfrak {i}}=o(1)$. We note that the right hand side is a Dirichlet polynomial of length

$$\begin{aligned} \leqslant \prod _{\mathfrak {i}=0}^JZ_\mathfrak {i}^{10\ell _\mathfrak {i}} = T^{\tfrac{1}{10}\sum _{\mathfrak {i}=0}^Je^{-\mathfrak {i}/4}} \leqslant T^{1/2} \end{aligned}$$

(6.8)

We can now state an upper bound for the $|M(\tfrac{1}{2}+it)|$ in terms of these short Dirichlet polynomials.

Lemma 11

Assume GRH for $L(s,\pi _j)$ for $1\leqslant j\leqslant m$ and let $t\in [T,2T]$. Then either

$$\begin{aligned} |{P}_{0,\mathfrak {j}}(t)|> \ell _0 \end{aligned}$$

for some $0\leqslant \mathfrak {j}\leqslant J$ or

$$\begin{aligned} |M(\tfrac{1}{2}+it)|^{2}&\ll \exp \bigg (\frac{1}{2}\sqrt{\frac{\log T}{\log _2 T\log _3 T}}\bigg )\prod _{\mathfrak {i}=0}^J \big |{\mathcal {N}}_{\mathfrak {i},J}(t)\big |^2\\&\quad + (\log T)^{O(1)}\sum _{\begin{array}{c} 0\leqslant \mathfrak {j}\leqslant J-1\\ \mathfrak {j}+1\leqslant l\leqslant J \end{array}} \exp \Big (\frac{C_M\log T}{\log Z_{\mathfrak {j}}}\Big ) \bigg (\frac{|{P}_{\mathfrak {j}+1,l}(t)|}{\ell _{\mathfrak {j}+1}} \bigg )^{2r_\mathfrak {j}}\prod _{\mathfrak {i}=0}^\mathfrak {j} \big |{\mathcal {N}}_{\mathfrak {i},\mathfrak {j}}(t)\big |^2. \end{aligned}$$

Proof

Suppose $|{P}_{0,\mathfrak {j}}(t)|<\ell _0$. For $0\leqslant \mathfrak { j}\leqslant J-1$ let

$$\begin{aligned} S(j)=\left\{ t\in [T,2T]: \begin{array}{ll} |{P}_{\mathfrak {i},l}(t)|\leqslant \ell _\mathfrak {i} &{}\quad \forall 1\leqslant \mathfrak {i}\leqslant \mathfrak {j},\,\, \forall \mathfrak {j}\leqslant l\leqslant J;\\ |{P}_{\mathfrak {j}+1,l}(t)|>\ell _{\mathfrak {j}+1} &{}\quad \text { for some } \mathfrak {j}+1\leqslant l\leqslant J \end{array} \right\} \end{aligned}$$

and

$$\begin{aligned} S(J)=\bigg \{t\in [T,2T]: |{P}_{\mathfrak {i},J}(t)|\leqslant \ell _\mathfrak {i}\qquad \forall 1\leqslant \mathfrak {i}\leqslant J\bigg \}. \end{aligned}$$

Then since $[T,2T]=\cup _{\mathfrak {j}=0}^J S(\mathfrak {j})$, for $t\in [T,2T]$ we have

$$\begin{aligned} \left| M\left( \tfrac{1}{2}+it\right) \right| ^{2} \leqslant \mathbbm {1}_{t\in S(J)}\cdot \left| M\left( \tfrac{1}{2}+it\right) \right| ^{2}+\sum _{\begin{array}{c} 0\leqslant \mathfrak {j}\leqslant J-1\\ \mathfrak {j}+1\leqslant l\leqslant J \end{array}}\mathbbm {1}_{t\in S_l(\mathfrak {j})}\cdot \left| M\left( \tfrac{1}{2}+it\right) \right| ^{2}\nonumber \\ \end{aligned}$$

(6.9)

where

$$\begin{aligned} S_l(\mathfrak {j})=\left\{ t\in [T,2T]: \begin{array}{ll} |{P}_{\mathfrak {i},l}(t)|\leqslant \ell _\mathfrak {i} &{} \quad \forall 1\leqslant \mathfrak {i}\leqslant \mathfrak {j},\,\, \forall \mathfrak {j}\leqslant l\leqslant J;\\ |{P}_{\mathfrak {j}+1,l}(t)|>\ell _{\mathfrak {j}+1} &{} \end{array} \right\} . \end{aligned}$$

We apply Corollary 9 to each M on the right hand side of (6.9). If $t\in S_l(\mathfrak {j})$ then we take $Z=Z_\mathfrak {j}$ to give

$$\begin{aligned} \left| M\left( \tfrac{1}{2}+it\right) \right| ^{2} \ll \exp \bigg (2\Re \sum _{p\leqslant Z_\mathfrak {j}}\frac{b(p)w_\mathfrak {j}(p)}{p^{1/2+it}} +\frac{C_M\log T}{\log Z_\mathfrak {j}} + O(\log _2 T)\bigg ). \end{aligned}$$

For the first sum over primes in the exponential we apply (6.7). To capture the small size of the set, we multiply by

$$\begin{aligned} \bigg (\frac{|{P}_{\mathfrak {j}+1,l}(t)|}{\ell _{\mathfrak {j}+1}}\bigg )^{2r_\mathfrak {j}}>1. \end{aligned}$$

If $t\in S(J)$ then we omit this last step. $\square$

6.4 Applying the inequality

We apply Lemma 11 to compute

$$\begin{aligned} \int _T^{2T}\left| M\left( \tfrac{1}{2}+it\right) \right| ^2|R(t)|^2dt \end{aligned}$$

By Lemma 10 we may disregard the t for which $|P_{0,\mathfrak {j}}(t)|>\ell _0$ since this gives a contribution o(T). By Lemma 11 the integral over the remaining set is then

$$\begin{aligned}{} & {} \ll \int _T^{2T}\bigg ( \exp \bigg (\frac{1}{2}\sqrt{\frac{\log T}{\log _2 T\log _3 T}}\bigg )\prod _{\mathfrak {i}=0}^J \big |{\mathcal {N}}_{\mathfrak {i},J}(t)\big |^2 + (\log T)^{O(1)}\nonumber \\{} & {} \quad \times \sum _{\begin{array}{c} 0\leqslant \mathfrak {j}\leqslant J-1\\ \mathfrak {j}+1\leqslant l\leqslant J \end{array}} \exp \Big (\frac{C_M\log T}{\log Z_\mathfrak {j}}\Big )\bigg (\frac{|{P}_{\mathfrak {j}+1,l}(t)|}{\ell _{\mathfrak {j}+1}}\bigg )^{2r_\mathfrak {j}}\prod _{\mathfrak {i}=0}^\mathfrak {j} \big |{\mathcal {N}}_{\mathfrak {i},\mathfrak {j}}(t)\big |^2\bigg )|R(t)|^2dt.\nonumber \\ \end{aligned}$$

(6.10)

To facilitate computations we note the following general observations. Suppose we are given R sets $\mathcal {S}_j\subset \mathbb {N}$ and Dirichlet polynomials

$$\begin{aligned} A_j(s) = \sum _{n\in {{\mathcal {S}}}_j} a_j(n) n^{-s}, \end{aligned}$$

where the $\prod _{j=1}^{R} n_j\leqslant M=o(T)$ for all $n_j \in {{\mathcal {S}}}_j$. Then by the mean value theorems for Dirichlet polynomials we have

$$\begin{aligned} \frac{1}{T}\int _T^{2T}\prod _{j=1}^R \big |A_j(it)\big |^2dt \sim&\sum _{n\leqslant M}\Big |\sum _{\begin{array}{c} n=n_1\cdots n_R\\ n_j\in \mathcal {S}_j \end{array}}a_1(n_1)\cdots a_R(n_R)\Big |^2 \end{aligned}$$

If for any $j_1,j_2$ with $j_1\ne j_2$ the elements of $\mathcal {S}_{j_1}$ are all coprime to the elements of $\mathcal {S}_{j_2}$ then there is at most one way to write $n= \prod _{j=1}^{R} n_j$ with $n_j \in {{\mathcal {S}}}_j$ and so

$$\begin{aligned} \frac{1}{T} \int _T^{2T} \prod _{j=1}^{R} |A_j(it)|^2 dt&= (1+O(NT^{-1})) \sum _{n\le N} \Big | \sum _{\begin{array}{c} n= n_1 \cdots n_R \\ n_j\in {{\mathcal {S}}}_j \end{array} }\prod _{j=1}^{R} a_j(n_j) \Big |^2 \\&= (1+O(NT^{-1})) \prod _{j=1}^R \Big ( \sum _{n_j \in {{\mathcal {S}}}_j} |a_j(n_j)|^2 \Big ) \\&= (1+ O(NT^{-1} ))^{1-R} \prod _{j=1}^{R} \Big ( \frac{1}{T} \int _{T}^{2T} |A_j(it)|^2 dt \Big ). \end{aligned}$$

Since $\prod _{\mathfrak {i}=0}^JN_{\mathfrak {i}, \mathfrak {j}}(t)$ is a Dirichlet polynomial of length $\leqslant T^{1/2}$ by (6.8) and R(t) is a Dirichlet polynomial of length $X=T^\Delta$ with $\Delta <1/2$, we can apply the above observations so that (6.10) becomes

$$\begin{aligned}{} & {} \ll T\exp \bigg (\frac{1}{2}\sqrt{\frac{\log T}{\log _2 T\log _3 T}}\bigg ) \frac{1}{T}\int _{T}^{2T} |\mathcal {N}_{0,J}(t)|^2|R(t)|^2dt \cdot \prod _{\mathfrak {i}=1}^J \frac{1}{T}\int _{T}^{2T}\big |{\mathcal {N}}_{\mathfrak {i},J}(t)\big |^2dt\\{} & {} \quad + T(\log T)^{O(1)} \sum _{\begin{array}{c} 0\leqslant \mathfrak {j}\leqslant J-1\\ \mathfrak {j}+1\leqslant l\leqslant J \end{array}} \exp \Big (\frac{C_M\log T}{\log Z_\mathfrak {j}}\Big ) \frac{1}{T}\int _{T}^{2T} |\mathcal {N}_{0,\mathfrak {j}}(t)|^2|R(t)|^2dt\\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \prod _{\mathfrak {i}=1}^\mathfrak {j} \frac{1}{T}\int _{T}^{2T}\big |{\mathcal {N}}_{\mathfrak {i},\mathfrak {j}}(t)\big |^2dt \frac{1}{T}\int _{T}^{2T} \bigg (\frac{|{P}_{\mathfrak {j}+1,l}(t)|}{\ell _{\mathfrak {j}+1}}\bigg )^{2r_\mathfrak {j}} dt. \end{aligned}$$

Note that, by (6.3) and (6.4) we have

$$\begin{aligned} \exp \Big (\frac{C_M\log T}{\log Z_\mathfrak {j}}\Big ) \cdot \frac{1}{T}\int _T^{2T} \bigg (\frac{|{P}_{\mathfrak {j}+1,l}(t)|}{\ell _{\mathfrak {j}+1}}\bigg )^{2r_\mathfrak {j}} dt&\ll \exp \Big (-\frac{C\log T}{e^{\mathfrak {j}+1}(\log _2 T)^{1+\epsilon }}\Big )\\&\ll \exp \big (-(\log T)^{1/2+o(1)}\big ). \end{aligned}$$

Since the number of terms in the sum over $\mathfrak {j},l$ along with the $(\log T)^{O(1)}$ term can be absorbed into this exponential, we arrive at

$$\begin{aligned}{} & {} \frac{1}{T}\int _T^{2T}|M(\tfrac{1}{2}+it)|^2|R(t)|^2dt\nonumber \\{} & {} \quad \ll \exp \bigg (\sqrt{\frac{\log T}{\log _2 T\log _3 T}}\bigg ) \max _{\mathfrak {j}\leqslant J} \frac{1}{T}\int _T^{2T} |\mathcal {N}_{0,\mathfrak {j}}(t)|^2|R(t)|^2dt \prod _{\mathfrak {i}=1}^\mathfrak {j} \frac{1}{T}\int _T^{2T}\big |{\mathcal {N}}_{\mathfrak {i},\mathfrak {j}}(t)\big |^2dt.\nonumber \\ \end{aligned}$$

(6.11)

6.5 Computing the mean values

It remains to compute

$$\begin{aligned} \frac{1}{T}\int _T^{2T} |\mathcal {N}_{0,\mathfrak {j}}(t)|^2|R(t)|^2dt \cdot \prod _{\mathfrak {i}=1}^\mathfrak {j} \frac{1}{T}\int _T^{2T}\big |{\mathcal {N}}_{\mathfrak {i},\mathfrak {j}}(t)\big |^2dt. \end{aligned}$$

Applying the mean value theorem for Dirichlet polynomials we have

$$\begin{aligned} \frac{1}{T}\int _T^{2T}|\mathcal {N}_{0,\mathfrak {j}}(t)|^2|R(t)|^2dt = (1+O(T^{-\epsilon })) \sum _{\begin{array}{c} m_1n_2=m_2n_2\\ \Omega (m_j)\leqslant \ell _0\\ p|m_j\implies p\leqslant Z_0\\ n_j\leqslant X \end{array}} \frac{\mathfrak {c}_\mathfrak {j}(m_1)\overline{\mathfrak {c}_\mathfrak {j}(m_2)}r(n_1)\overline{r(n_2)}}{(m_1m_2)^{1/2}} \end{aligned}$$

and

$$\begin{aligned} \prod _{\mathfrak {i}=1}^\mathfrak {j}\frac{1}{T}\int _T^{2T}|\mathcal {N}_{\mathfrak {i},\mathfrak {j}}(t)|^2dt = (1+O(T^{-\epsilon }))\prod _{\mathfrak {i}=1}^\mathfrak {j} \sum _{\begin{array}{c} \Omega (m)\leqslant \ell _\mathfrak {i}\\ p|m\implies Z_{\mathfrak {i}-1}<p\leqslant Z_\mathfrak {i} \end{array}} \frac{|\mathfrak {c}_\mathfrak {j}(m)|^2}{m} \end{aligned}$$

with

$$\begin{aligned} \mathfrak {c}_\mathfrak {j}(m)=c(m)\mathfrak {g}(m)W_\mathfrak {j}(m) \end{aligned}$$

where we recall the definition of these coefficients from (6.5), (6.6).

Now,

$$\begin{aligned} \prod _{\mathfrak {i}=1}^\mathfrak {j} \sum _{\begin{array}{c} p|n\implies Z_{\mathfrak {i}-1}<p\leqslant Z_\mathfrak {i} \end{array}} \frac{|\mathfrak {c}_\mathfrak {j}(m)|^2}{m} \leqslant \exp \bigg (\sum _{Z_0<p\leqslant Z_\mathfrak {j}}\frac{|b(p)|^2}{p}\bigg ) \ll \bigg (\frac{\log Z_J}{\log Z_0}\bigg )^{m-1}\nonumber \\ \end{aligned}$$

(6.12)

by (6.1). It thus suffices to show

$$\begin{aligned} \begin{aligned}&\sum _{\begin{array}{c} m_1n_2=m_2n_2\\ \Omega (m_j)\leqslant \ell _0\\ p|m_j\implies p\leqslant Z_0\\ n_j\leqslant X \end{array}} \frac{\mathfrak {c}_\mathfrak {j}(m_1)\overline{\mathfrak {c}_\mathfrak {j}(m_2)}r(n_1)\overline{r(n_2)}}{(m_1m_2)^{1/2}}\\&\quad \ll \prod _p\bigg (1+|r(p)|^2+2(1+o(1))\Re \frac{r(p)\overline{b(p)}}{p^{1/2}}\bigg ). \end{aligned} \end{aligned}$$

(6.13)

Assuming this for the moment, plugging (6.12) and (6.13) into (6.11) gives

$$\begin{aligned}{} & {} \frac{1}{T}\int _T^{2T}|M(\tfrac{1}{2}+it)|^2|R(t)|^2dt\\{} & {} \quad \ll \exp \bigg (\sqrt{\frac{\log T}{\log _2 T\log _3 T}}\bigg ) \prod _p\bigg (1+|r(p)|^2+2(1+o(1))\Re \frac{r(p)\overline{b(p)}}{p^{1/2}}\bigg ). \end{aligned}$$

and Proposition 4 follows.

To prove (6.13) we apply Rankin’s trick to find that the sum on the left there is for any $\alpha >0$,

$$\begin{aligned}&\sum _{\begin{array}{c} m_1n_2=m_2n_2\\ p|n\implies p\leqslant Z_0 \end{array}} \frac{\mathfrak {c}_\mathfrak {j}(m_1)\overline{\mathfrak {c}_\mathfrak {j}(m_2)}r(n_1)\overline{r(n_2)}}{(m_1m_2)^{1/2}}\nonumber \\&\quad + O\bigg (e^{-\ell _0}\sum _{\begin{array}{c} m_1n_2=m_2n_2\\ p|m_j\implies p\leqslant Z_0 \end{array}} \frac{|\mathfrak {c}_\mathfrak {j}(m_1){\mathfrak {c}_\mathfrak {j}(m_2)}r(n_1){r(n_2)} |e^{\Omega (m_1)}}{(m_1m_2)^{1/2}} \bigg )\nonumber \\&\quad + O\bigg ( X^{-\alpha }\sum _{\begin{array}{c} m_1n_2=m_2n_2\\ p|m_j\implies p\leqslant Z_0 \end{array}} \frac{|\mathfrak {c}_\mathfrak {j}(m_1){\mathfrak {c}_\mathfrak {j}(m_2)}r(n_1){r(n_2)}|n_1^\alpha }{(m_1m_2)^{1/2}}\bigg ) \end{aligned}$$

(6.14)

by symmetry.

The main term here is

$$\begin{aligned}&\prod _{p\leqslant Z_0}\sum _{m_1+n_1=m_2+n_2}\frac{b(p)^{m_1}\overline{b(p)^{m_2}} w_\mathfrak {j}(p)^{m_1}w_\mathfrak {j}(p)^{m_2}r(p^{n_1})\overline{r(p^{n_2})}}{m_1!m_2!p^{(m_1+m_2)/2}}\\&\quad = \prod _{p\leqslant Z_0}\bigg ((1+|r(p)|^2)\sum _{m\geqslant 0}\frac{|b(p)|^{2m}w_\mathfrak {j}(p)^{2m}}{m!^2p^m}\\&\qquad + 2\Re \frac{r(p)\overline{b(p)}}{p^{1/2}}\sum _{m\geqslant 0}\frac{|b(p)|^{2m}w_\mathfrak {j}(p)^{2m}}{m!(m+1)!p^m} \bigg )\\&\quad =\mathcal {K}(X)\prod _{p}\bigg (1+|r(p)|^2 +2\Re \frac{r(p)\overline{b(p)}}{p^{1/2}}B(p)\bigg ) \end{aligned}$$

where

$$\begin{aligned} \mathcal {K}(X)=\prod _{p\leqslant Z_0}\sum _ {m\geqslant 0}\frac{|b(p)|^{2m}w_\mathfrak {j}(p)^{2m}}{m!^2p^m} \ll \exp \bigg (\sum _{p\leqslant Z_0}\frac{|b(p)|^2}{p}\bigg )\ll (\log Z_0)^{m-1}, \end{aligned}$$

by (6.1) and

$$\begin{aligned} B(p)=\frac{\sum _{m\geqslant 0}{|b(p)|^{2m}w_\mathfrak {j}(p)^{2m}}/{m!^2p^m} }{\sum _{m\geqslant 0}{|b(p)|^{2m}w_\mathfrak {j}(p)^{2m}}/{m!(m+1)!p^m}}=1+O\Big (\frac{1}{p^{2/(\max d_i^2+1)}}\Big ) \end{aligned}$$

by (2.4). Since $B(p)=1+o(1)$ for p in the support of r and $(\log Z_0)^{m-1}$ can be absorbed into this o(1) term of the exponential, it remains to show that the error terms of (6.14) are of a lower order than this.

With a similar calculation the first error term there is

$$\begin{aligned} \ll e^{-\ell _0}(\log Z_0)^{e(m-1)}\prod _{p}\bigg (1+|r(p)|^2+2e(1+o(1))\frac{|r(p)b(p)|}{p^{1/2}}\bigg ). \end{aligned}$$

Since $|r(p)|^2, r(p)b(p)=o(1)$ in the support of $r(\cdot )$ as in (3.2), the ratio of this to the main term is then

$$\begin{aligned} \ll e^{-\ell _0}(\log Z_0)^{e(m-1)}\exp \bigg (4e\sum _{p\leqslant Z_0}\frac{|r(p)b(p)|}{p^{1/2}}\bigg )=o(1) \end{aligned}$$

on recalling that $\ell _0=\log T/100(\log \mathcal {L})^2\asymp \log T/(\log _2 T)^2$ and noting that the sum in the exponential is $\ll \sqrt{\log T/\log _2 T}$.

The second error term is

$$\begin{aligned} \ll X^{-\alpha }(\log Z_0)^{m-1}\prod _{p}\bigg (1+|r(p) |^2p^\alpha +2(1+o(1))\frac{|r(p)b(p)|}{p^{1/2-\alpha }}\bigg ). \end{aligned}$$

The ratio of this to the main term is

$$\begin{aligned} \ll \exp \bigg (-\alpha \log X+\sum _{\mathcal {L}^2<p\leqslant \exp ((\log \mathcal {L})^2)}|a(p)|^2\frac{\mathcal {L}^2}{p\log ^2 p}(p^{\alpha }-1)+O\Big (\sqrt{\frac{\log T}{\log _2 T}}\Big )\bigg ) \end{aligned}$$

for $\alpha =1/(\log \mathcal {L})^3$. The usual computations, as in (4.6), show this is o(1).

7 Simultaneous extreme values of twists of GL(2) cusp forms: Proof of Theorem 2

Let f, g be a fixed primitive (holomorphic or Maaß) cusp forms with respect to $\Gamma _0(r)$ and $\Gamma _0(r')$ with trivial central character. As before, if we can find $R(\chi )$ and V such that

$$\begin{aligned}&\frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q} |L(1/2,f\otimes \chi )\overline{ L(1/2,g\otimes \chi )}R(\chi )|^2\nonumber \\&\quad \geqslant \frac{V}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q} \Big (|L(1/2,f\otimes \chi )R(\chi )|^2+|L(1/2,g\otimes {{\bar{\chi }}})R(\chi )|^2\Big ) \end{aligned}$$

(7.1)

with $*$ meaning the sum is over primitive characters modulo q and $\phi ^*(q)=q-2$, then we must have

$$\begin{aligned} \max _{\chi \bmod q} \min _{f,g}\Big (|L(1/2,f\otimes \chi )|, L(1/2,g\otimes \chi )|\Big )\geqslant \sqrt{V}. \end{aligned}$$

To estimate these mean values we follow [5] quite closely and so retain some of their methods, notation and set-up for ease of comparison, although it may differ from our previous sections slightly.

We consider the sum on the left hand side of (7.1) first. From Cauchy’s inequality we have

$$\begin{aligned}&\frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q}|L(1/2,f\otimes \chi )\overline{L(1/2,g\otimes \chi )}R(\chi )|^2\\&\quad \geqslant \Big (\frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q} |R(\chi )|^2\Big )^{-1}\Big ( \frac{1}{\phi ^*(q)} \mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q} L(1/2,f\otimes \chi )\overline{L(1/2,g\otimes \chi )}|R(\chi )|^2\Big )^2 \end{aligned}$$

Thus it make sense to choose $R(\chi )$ such that $|L(1/2,f\otimes \chi )L(1/2,g\otimes \chi )|$ is large and we follow the choice of $R(\chi )$ in [5, Sect. 7.5.1].

Let $\lambda _f^*, \lambda _g^*$ be multiplicative functions supported on squarefree positive integers defined by $\lambda _f^*(p)=(1-1/p)^{-1}(\lambda _f(p)-\lambda _g(p)/p)$ and $\lambda _g^*(p)=(1-1/p)^{-1}(\lambda _g(p)-\lambda _f(p)/p)$. For $u\geqslant 1$ some parameter depending only on f and g, let

$$\begin{aligned} {\mathcal {G}}:=\{n\geqslant 1: (n, urr')=1, \lambda _f^*(n)\lambda _g^*(n)\not =0, {\text {sgn}}(\lambda _f^*(n))={\text {sgn}}(\lambda _g^*(n))\}. \end{aligned}$$

(7.2)

Define

$$\begin{aligned} \varpi (p)={\left\{ \begin{array}{ll} \lambda _f^*(p)\lambda _g^*(p)(\lambda _f^*(p)+\lambda _g^*(p)), &{}\quad p\in {\mathcal {G}},\\ 0, &{} \quad p\not \in {\mathcal {G}}. \end{array}\right. } \end{aligned}$$

Let

$$\begin{aligned} \omega (n)&=|\varpi (n)|^2,\ \omega _1'(n)=\varpi (n)\lambda _f^*(n), \\ \omega _2'(n)&=\varpi (n)\lambda _g^*(n),\ \omega '(p)=\omega _1'(p)+\omega _2'(p) \end{aligned}$$

and

$$\begin{aligned} R(\chi )=\sum _{n\leqslant N}r(n)\varpi (n)\chi (n) \end{aligned}$$

where

$$\begin{aligned} r(p) = {\left\{ \begin{array}{ll} \frac{\mathcal {L}}{p^{1/2}\log p}{} &{}\quad \text {for } \mathcal {L}^2\leqslant p \leqslant \exp ((\log \mathcal {L})^2)\\ 0 &{} \quad \text {otherwise } \end{array}\right. } \end{aligned}$$

(7.3)

and

$$\begin{aligned} \mathcal {L}=\sqrt{a_{\omega }^{-1}\log N\log \log N} \end{aligned}$$

for some constant $a_\omega$ as in [5, Eq. (7.55)]

Fixing an arbitrary $\delta >0$ and $N\leqslant q^{1/360-\delta }$ we have that using [5, Lemmas 7.19, 7.10]

$$\begin{aligned} \frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q}|R(\chi )|^2\sim \prod _{p}\Big (1+r(p)^2\omega (p)\Big ) \end{aligned}$$

and using [5, Lemmas 7.19, 7.12, 7.14] there exists a squarefree integer $u\ge 1$ coprime to $rr'$ such that

$$\begin{aligned}&\frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q}|R(\chi )|^2 L(1/2,f\otimes \chi )L(1/2,g\otimes {\bar{\chi }})\chi (u)\\&\quad =L^*(1,f\otimes g)(\nu +o(1))\prod _{p}\Big (1+r(p)^2 \omega (p)+\frac{r(p)\omega '(p)}{\sqrt{p}}\Big )\\&\qquad +O\Big (q^{-\delta }\prod _{p}\big (1+r(p)^2\omega (p)\big )\Big ), \end{aligned}$$

where $L^*(s,f\otimes g)$ is as in [5, Eq. (2.7)], $\nu \not =0$ is a constant depending on f, g only. As in [5], the $\chi (u)$ is introduced to break the symmetry of $\chi$ and $\overline{\chi }$. Since $|\chi (u)|\leqslant 1$ and $L^*(f\otimes g, 1)\not =0$ ( [5, Lemma 2.6]) we see that

$$\begin{aligned}&\frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q}|L(1/2,f\otimes \chi )\overline{L(1/2,g\otimes \chi )}R(\chi )|^2 \nonumber \\&\quad \gg _{f,g} \prod _{p}\Big (1+r(p)^2 \omega (p)+\frac{r(p)w'(p)}{\sqrt{p}}\Big )^2 \Big (1+r(p)^2 \omega (p)\Big )^{-1}. \end{aligned}$$

(7.4)

We now turn to getting an upper bound for the mean squares on the right of (7.1). Similarly to the proof of [5, Lemma 7.9], we have for $(\ell , \ell ')=(\ell \ell ', qrr')=1$, $\ell ,\ell '\leqslant L$,

$$\begin{aligned}{} & {} \frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q} |L(1/2,f\otimes \chi )|^2\chi (\ell ) \chi (\ell ')\\{} & {} \quad =\frac{1}{2}{\text {MT}}^{+}(f,f;\ell , \ell ') +\frac{1}{2}{\text {MT}}^{-}(f,f;\ell , \ell ') +O(L^{3/2}q^{-1/144+\epsilon }), \end{aligned}$$

where

$$\begin{aligned} {\text {MT}}^{\pm }(f,f;\ell , \ell ')&=\frac{1}{2\pi i}\int _{(2)}\frac{ L_\infty ^\pm (f, \pm , \frac{1}{2}+ u)}{L_\infty ^2(f, \pm ,\frac{1}{2})}\frac{D(1+2u;\ell ,\ell ')}{(\ell \ell ')^{1/2+u}}G(u)(q^2|rr'|)^u\frac{du}{u},\\ D(s; \ell , \ell ')&=\sum _{n}\frac{\lambda _f(\ell n)\lambda (\ell 'n)}{n^s}\\&=L^*(f\otimes f, s)\prod _{p\mid \ell \ell '}(1-p^{-2s})^{-1}\prod _{p\mid \ell \ell '}\left( \lambda _f(p)-\frac{\lambda _f(p)}{p^s}\right) . \end{aligned}$$

Here $G(u)=\cos (\frac{\pi u}{4A})^{-16 A}$ for some $A\ge 2$ and $L_\infty (f, \pm , s)=L_\infty (f\otimes \chi , s)$ for $\chi (-1)=\pm 1$ (see [5, Lemma 2.1] for definitions of $L_\infty (f\otimes \chi , s)$). Shifting the contour of integration to $\Re (u)=-\frac{1}{4}+\epsilon$, we encounter a double pole at $u=0$ so that

$$\begin{aligned} {\text {MT}}^\pm&= \frac{\lambda _f^*(\ell \ell ')}{\sqrt{\ell \ell '}}\Big (\frac{L^*({\text {Sym}}^2f, 1)}{\prod _{p\mid r}(1+p^{-1})\zeta (2)}+2\log (|r|q)-\log (\ell \ell ')\\&\qquad \qquad \quad \qquad +C_f+\sum _{p\mid \ell \ell '}\frac{2\log p}{p+1}\Big ) \end{aligned}$$

using [5, Eq. (2.9)] for ${\text {Res}}_{s=1}L^*(f\otimes f, 1)$ and

$$\begin{aligned} C_f=\frac{d}{du}\frac{L_\infty ^\pm (f, \pm , \frac{1}{2}+u)}{L^2_\infty (f, \pm , 1/2)}\Big \vert _{u=0}. \end{aligned}$$

Therefore, we have that

$$\begin{aligned}{} & {} \frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q}|L(1/2,f\otimes \chi )R(\chi )|^2 \\{} & {} \quad =\sum _{d}|r(d)|^2 \omega (d)\sum _{\begin{array}{c} \ell , \ell '\le N/d\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\varpi (\ell \ell ')}{\sqrt{\ell \ell '}}\lambda _f^*(\ell \ell ')\\{} & {} \qquad \times \Big (A_f+2\log q-\log (\ell \ell ')+\sum _{p\mid \ell \ell '} \frac{2\log p}{p+1}+O(N^{3/2}q^{-1/144})\Big ) \end{aligned}$$

where $A_f=\frac{L^*({\text {Sym}}^2f, 1)}{\prod _{p\mid r}(1+p^{-1})\zeta (2)}+C_f+2\log |r|$. It follows that

$$\begin{aligned}&\frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q}|L(1/2,f\otimes \chi )R(\chi )|^2 +O(N^{7/2}q^{-144}) \\&\quad =\sum _{d}|r(d)|^2 \omega (d)\sum _{\begin{array}{c} \ell , \ell '\le N/d\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\varpi (\ell \ell ')}{\sqrt{\ell \ell '}}\Big (A_f\lambda _f^*(\ell \ell ') +\lambda _f^*(\ell \ell ') \big (2\log q-\log (\ell \ell ') \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +2\sum _{p\mid \ell \ell '}\frac{\log p}{p+1}\big )\Big ) \end{aligned}$$

Note from the support of $\varpi$ in (7.2), we have $0\leqslant \varpi (p)\lambda _f^*(p)=\omega _1'(p)$ and thus

$$\begin{aligned} \sum _{\begin{array}{c} \ell , \ell '\le N/d\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\varpi (\ell \ell ')\lambda _f^*(\ell \ell ')}{\sqrt{\ell \ell '}}&\le \prod _{\begin{array}{c} \ell , \ell '\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\varpi (\ell \ell ') \lambda _f^*(\ell \ell ')}{\sqrt{\ell \ell '}}\\&=\prod _{\begin{array}{c} \ell , \ell '\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\omega _1'(\ell \ell ')}{\sqrt{\ell \ell '}}. \end{aligned}$$

Therefore

$$\begin{aligned}&\sum _{d}|r(d)|^2 \omega (d)\sum _{\begin{array}{c} \ell , \ell '\le N/d\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\varpi (\ell \ell ')}{\sqrt{\ell \ell '}}A_f\lambda _f^*(\ell \ell ')\\&\quad \ll _f \prod _{p}\Big (1+r(p)^2 \omega (p)+\frac{2r(p)\omega _1'(p)}{\sqrt{p}}\Big ). \end{aligned}$$

With the current choice of $\varpi , N$, we also have that

$$\begin{aligned} 2\log q-\log (\ell \ell ')+\sum _{p\mid \ell \ell '}\frac{2\log p}{p+1}\geqslant 0, \end{aligned}$$

which together with $\varpi (\ell )\lambda _f^*(\ell )\ge 0$ gives

$$\begin{aligned}&\sum _{d}|r(d)|^2 \omega (d)\sum _{\begin{array}{c} \ell , \ell '\le N/d\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\varpi (\ell \ell ')}{\sqrt{\ell \ell '}}\Big (2\log q-\log (\ell \ell ')+\sum _{p\mid \ell \ell '}\frac{2\log p}{p+1}\Big )\lambda _f^*(\ell \ell ')\\&\quad \ll \log q\sum _{d}|r(d)|^2 \omega (d)\sum _{\begin{array}{c} \ell , \ell '\\ (\ell \ell ',d)=1 \end{array}}\frac{r(\ell \ell ')\varpi (\ell \ell ')}{\sqrt{\ell \ell '}}\lambda _f^*(\ell \ell ')\\&\quad \ll \ \log q\prod _{p}\Big (1+r(p)^2\omega (p)+\frac{2r(p)\omega _1'(p)}{\sqrt{p}}\Big ). \end{aligned}$$

Therefore, we have

$$\begin{aligned}&\frac{1}{\phi ^*(q)}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\chi \bmod q} \Big (|L(1/2,f\otimes \chi )R(\chi )|^2+ |L(1/2,g\otimes \chi )R(\chi )|^2\Big )\nonumber \\&\quad \ll \max _{i=1,2}\ \log q\prod _{p}\Big (1+r(p)^2 \omega (p)+\frac{2r(p)\omega _i'(p)}{\sqrt{p}}\Big ). \end{aligned}$$

(7.5)

Combining (7.4) and (7.5), we can choose

$$\begin{aligned} V= & {} \min _{i=1,2}\frac{1}{\log q}\prod _{p}\Big (1+r(p)^2 \omega (p) +\frac{r(p)\omega '(p)}{\sqrt{p}}\Big )^2(1+r(p)^2 \omega (p))^{-1}\\{} & {} \times \Big (1+r(p)^2 \omega (p)+\frac{2r(p)\omega _i'(p)}{\sqrt{p}}\Big )^{-1}. \end{aligned}$$

We have from [5, Proof of Lemma 7.5] that

$$\begin{aligned}&\log \prod _{p}\Big (1+r(p)^2 \omega (p) +\frac{r(p)\omega '(p)}{\sqrt{p}} \Big )^2(1+r(p)^2 \omega (p))^{-1}\\&\qquad \times \Big (1+r(p)^2 \omega (p)+\frac{2r(p) \omega _i'(p)}{\sqrt{p}}\Big )^{-1}\\&\quad =\log \prod _{p}\Big (1 +\frac{r(p)\omega '(p)}{\sqrt{p}(1+r(p)^2 \omega (p))}\Big )^2\Big (1+\frac{2r(p)\omega _i'(p)}{\sqrt{p}(1+r(p)^2\omega (p))}\Big )^{-1}\\&\quad =\mathcal {L}\sum _{L^2 \le p\le \exp (\log ^2 L)}\frac{2\omega '(p) -2\omega _i'(p)}{p\log p}+O_{\omega , \omega ', \delta }\Big (\frac{L}{(\log L)^{1+\delta }}\Big ) \end{aligned}$$

Since $\lambda _f^*(p)\lambda _g^*(p)(\lambda _f^*(p)+\lambda _g^*(p))\lambda _f^*(p), \lambda _f^*(p)\lambda _g^*(p)(\lambda _f^*(p)+\lambda _g^*(p))\lambda _g^*(p)\leqslant 0$ when $p\not \in {\mathcal {G}}$, we have

$$\begin{aligned}&\sum _{L^2\le p\le \exp (\log ^2 L)} \frac{2\omega '(p) -2\omega _1'(p)}{p\log p}\\&\quad \geqslant \sum _{\begin{array}{c} \mathcal {L}^2 \leqslant p\leqslant \exp (\log ^2 \mathcal {L})\\ p\in {\mathcal {G}} \end{array}} \frac{2\lambda _f^*(p)\lambda _g^*(p)(\lambda _f^*(p)+\lambda _g^*(p))\lambda _g^*(p)}{p\log p}\\&\quad \geqslant \sum _{\begin{array}{c} \mathcal {L}^2 \le p\le \exp (\log ^2 \mathcal {L}) \end{array}} \frac{2\lambda _f^*(p)\lambda _g^*(p)(\lambda _f^*(p) +\lambda _g^*(p))\lambda _g^*(p)}{p\log p}\\&\quad = \frac{c}{\log L}+O\left( \frac{1}{(\log L)^2}\right) . \end{aligned}$$

for some positive $c=2n_{2,2}+2n_{1,3}$ using the notation for $n_{i,j}$ in [5, Corollary 2.17]. Thus we see that for every prime q sufficiently large depending on f, g, there exists a non-trivial character $\chi \bmod q$ such that

$$\begin{aligned} \min _{f,g }(|L(1/2,f\otimes \chi )|, |L(1/2,g\otimes \chi )|) \geqslant \sqrt{V}\geqslant \exp \bigg (c_{f,g}\sqrt{\frac{\log q}{\log \log q}}\bigg ) \end{aligned}$$

for some positive $c_{f,g}$. With $N=q^{1/360-\delta }$, we can take the constant

$$\begin{aligned} c_{f,g}=\frac{1}{2}\left( \frac{1}{6\sqrt{10}}\sqrt{1-360\delta }+o(1) \right) \frac{n_{2,2}+\min (n_{1,3}, n_{3,1})}{(n_{4,2}+2n_{3,3}+n_{2,4})^{1/2}}. \end{aligned}$$

In a generic situation, as in [5, Remark 7.20], i.e. where neither f nor g are of polyhedral type (in particular ${\text {Sym}}^{\textrm{k}}f, {\text {Sym}}^kg$ are cuspidal for all $k\le 4$) and if ${\text {Sym}}^k \pi _f\not \cong {\text {Sym}}^k \pi _g$ for $k\leqslant 4$, we see that

$$\begin{aligned} c_{f,g}=\frac{1}{12\sqrt{10}}+o(1). \end{aligned}$$

8 Simultaneous small values of quadratic twists: Proof of Theorem 4

Let f, g be holomorphic cusp forms of weight $\kappa \equiv 0 \bmod 4$ for $SL_2({\mathbb {Z}})$ and let $\chi _{d}(n)=\big (\frac{d}{n}\big )$ be the Kronecker symbol. Due to the non-negativity of $L(1/2,f\otimes \chi _{d})$, small values of $L(1/2,f\otimes \chi _{d})+L(1/2,g\otimes \chi _{d})$ implies simultaneous small values of $L(1/2, f\otimes \chi _{d})$ and $L(1/2, g\otimes \chi _{d})$. Let $\Phi (x):(0, \infty )\rightarrow {\mathbb {C}}$ be a smooth, compactly supported function. From [43, Theorem 1.4], we have for square-free $\ell$

$$\begin{aligned}&\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{(d,2)=1}\chi _{8d}(\ell )L(1/2,f\otimes \chi _{8d})\Phi (\frac{d}{X})\\&\quad =\frac{8X {\tilde{\Phi }}(1)}{\pi ^2 \sqrt{\ell }} L(1,{\text {Sym}}^2f)Z(1/2, \ell )+O(\ell ^{1/2+\epsilon }X^{1/2+\epsilon }) \end{aligned}$$

where the $*$ now denotes a sum over squarefree integers and

$$\begin{aligned}&L(1,{\text {Sym}}^2f)Z(1/2,\ell ) \\&\quad = \prod _{p\mid \ell }\frac{p^{3/2}}{2(p+1)}\Big (\Big (1-\frac{\lambda _f(p)}{\sqrt{p}}+\frac{1}{p}\Big )^{-1}-\Big (1+\frac{\lambda _f(p)}{\sqrt{p}}+\frac{1}{p}\Big )^{-1}\Big )\\&\qquad \times \prod _{p\not \mid 2\ell }\left( 1+\frac{p}{2(p+1)} \left( \left( 1-\frac{\lambda _f(p)}{\sqrt{p}}+\frac{1}{p}\right) ^{-1} +\left( 1+\frac{\lambda _f(p)}{\sqrt{p}}+\frac{1}{p}\right) ^{-1}-2\right) \right) . \end{aligned}$$

We write $L(1,{\text {Sym}}^2f)Z(1/2,\ell )=: L(1,{\text {Sym}}^2 f)Z(1/2, 1) \prod _{p\mid \ell }h_f(p)$ so that

$$\begin{aligned} h_f(p)&=\frac{p^{3/2}}{2(p+1)}\left( \left( 1-\frac{\lambda _f(p)}{\sqrt{p}} +\frac{1}{p}\right) ^{-1}-\left( 1+\frac{\lambda _f(p)}{\sqrt{p}}+\frac{1}{p}\right) ^{-1}\right) \\&\quad \times \left( 1+\frac{p}{2(p+1)}\left( \left( 1-\frac{\lambda _f(p)}{\sqrt{p}} +\frac{1}{p}\right) ^{-1}+\left( 1+\frac{\lambda _f(p)}{\sqrt{p}}+\frac{1}{p}\right) ^{-1}-2\right) \right) ^{-1}\\&=\lambda _f(p)+ O\big (\frac{|\lambda _f(p)|}{p}\big ). \end{aligned}$$

Thus for $R(\chi _{8d})=\sum _{n\le N}\mu (n)r(n)\varpi (n)\chi _{8d}(n)$ with $r, \varpi$ real multiplicative functions supported on squarefree integers

$$\begin{aligned}{} & {} \mathop {{\mathop {\sum }\nolimits ^*}}\limits _{(d, 2)=1}L( 1/2,f\otimes \chi _{8d})|R(\chi _{8d})|^2\Phi \left( \frac{d}{X}\right) \nonumber \\{} & {} \quad =\frac{8X{\tilde{\Phi }}(1)}{\pi ^2}L(1, {\text {Sym}}^2f)Z(1/2,1)\sum _{d\le N}r(d)^2\omega (d)\sum _{\begin{array}{c} n_1, n_2\le N/d\\ (n_1n_2,d)=1 \end{array}}\frac{\mu (n_1n_2)r(n_1n_2)\omega _1'(n_1n_2)}{\sqrt{n_1n_2}}\nonumber \\{} & {} \qquad +O( N^{5/2+\epsilon }X^{1/2+\epsilon }) \end{aligned}$$

(8.1)

where $\omega (n)=|\varpi (n)|^2$ and $\omega _1'(n)=\varpi (n)h_f(n)$. A similar expression holds when f is replaced by g. Now it remains to find r(n) and $\varpi (n)$.

As usual, let r(n) be multiplicative supported on squarefrees and satisfying (7.3) with $\mathcal {L}=\sqrt{a_{\omega }\log N\log \log N}$ where $a_\omega$ is defined as in [5, Eq. (7.2)]. Let $\tilde{{\mathcal {G}}}:=\{n\ge 1: h_f(n)h_g(n)\not =0, {\text {sgn}}(h_f(n))={\text {sgn}}(h_g(n))\}$ and define $\varpi$ as

$$\begin{aligned} \varpi (p)={\left\{ \begin{array}{ll} h_f(p)h_g(p)(h_f(p)+h_g(p)), &{} \quad p\in \tilde{{\mathcal {G}}},\\ 0, &{} \quad p\not \in \tilde{{\mathcal {G}}}. \end{array}\right. } \end{aligned}$$

Then similarly as before, we can evaluate the $d, n_1, n_2$-sum in (8.1) as

$$\begin{aligned} (1+o(1))\prod _{p}\Big (1+r(p)^2\omega (p)-\frac{2r(p)\omega _1'(p)}{\sqrt{p}}\Big ). \end{aligned}$$

With $N=X^{1/5-\delta }$ we see that (8.1) becomes

$$\begin{aligned}&\frac{8X{\tilde{\Phi }}(1)}{\pi ^2}L({\text {Sym}}^2f, 1/2)Z(1/2,1)(1+o(1))\prod _{p}\Big (1+r(p)^2 \omega (p)-\frac{2r(p)\omega _1'(p)}{\sqrt{p}}\Big )\\&\quad +O(X^{1-5\delta /2+\epsilon }). \end{aligned}$$

By standard computations (e.g. see [44]), we have

$$\begin{aligned} \mathop {{\mathop {\sum }\nolimits ^*}}\limits _{(d,2)=1}|R(\chi _{8d})|^2\Phi \left( \tfrac{d}{X}\right) \sim cX\prod _p\big (1+r(p)^2\omega (p)) \end{aligned}$$

for some positive constant c.

Thus, since $h_f(p)=\lambda _f(p)+O(p^{-1+\theta })$ with $\theta =7/64$, we have from [5, Corollary 2.17, Lemma 7.19, Proof of Lemma 7.5] that our ratio of mean values is

$$\begin{aligned} \ll \prod _{p}\Big (1-\frac{2r(p)\omega _1'(p)}{\sqrt{p}(1+r(p)^2\omega (p))}\Big ) \leqslant \exp \Big (- (n_{3,1}+n_{2,2}+o(1))\frac{\mathcal {L}}{2\log \mathcal {L}}\Big ) \end{aligned}$$

with $n_{i,j}$ defined as in [5, Corollary 2.17]. Therefore, we see that there exists d such that

$$\begin{aligned} \max (L(1/2,f\otimes \chi _{8d}), L(1/2,g\otimes \chi _{8d}))\leqslant \exp \Big (-{\tilde{c}}_{f,g}(\sqrt{\frac{1}{5}-\delta })\sqrt{\frac{\log X}{\log \log X}}\Big ). \end{aligned}$$

where positive ${\tilde{c}}_{f,g}=\frac{\min \big (n_{3,1}, n_{1,3}\big )+n_{2,2}}{\sqrt{a_\omega }}+o(1)>0$. In a generic situation, where neither f nor g are of polyhedral type (in particular ${\text {Sym}}^{\textrm{k}}f, {\text {Sym}}^kg$ are cuspidal for all $k\le 4$) and if ${\text {Sym}}^k \pi _f\not \cong {\text {Sym}}^k \pi _g$ for $k\le 4$, then ${\tilde{c}}_{f,g}=1+o(1)$ using [5, Eq. (7.55)] for $a_\omega$.