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Mathematische Zeitschrift

, Volume 292, Issue 3–4, pp 1105–1122 | Cite as

Standard zero-free regions for Rankin–Selberg L-functions via sieve theory

  • Peter HumphriesEmail author
  • Farrell Brumley
Open Access
Article

Abstract

We give a simple proof of a standard zero-free region in the t-aspect for the Rankin–Selberg L-function \(L(s,\pi \times \widetilde{\pi })\) for any unitary cuspidal automorphic representation \(\pi \) of \(\mathrm {GL}_n(\mathbb {A}_F)\) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.

Keywords

Cuspidal automorphic representation Rankin–Selberg Zero-free region 

Mathematics Subject Classification

11M26 (primary) 11F66 11N36 (secondary) 

1 Introduction

Let F be a number field, let n be a positive integer, and let \(\pi \) be a unitary cuspidal automorphic representation of \(\mathrm {GL}_n(\mathbb {A}_F)\) with L-function \(L(s,\pi )\), with \(\pi \) normalised such that its central character is trivial on the diagonally embedded copy of the positive reals. The proof of the prime number theorem due to de la Valleé–Poussin gives a zero-free region for the Riemann zeta function \(\zeta (s)\) of the form
$$\begin{aligned} \sigma > 1 - \frac{c}{\log (|t| + 3)} \end{aligned}$$
for \(s = \sigma + it\), and this generalises to a zero-free region for \(L(s,\pi )\) of the form
$$\begin{aligned} \sigma \ge 1 - \frac{c}{(n[F : \mathbb {Q}])^4 \log (\mathfrak {q}(\pi )(|t| + 3))} \end{aligned}$$
(1.1)
for some absolute constant \(c > 0\), where \(\mathfrak {q}(\pi )\) is the analytic conductor of \(\pi \) in the sense of [12, Equation (5.7)], with the possible exception of a simple real-zero \(\beta _{\pi } < 1\) when \(\pi \) is self-dual. A proof of this is given in [12, Theorem 5.10]; the method requires constructing an auxiliary L-function having a zero of higher order than the order of the pole at \(s = 1\), then using an effective version of Landau’s lemma [12, Lemma 5.9].

Now let \(\pi '\) be a unitary cuspidal automorphic representation of \(\mathrm {GL}_{n'}(\mathbb {A}_F)\), and consider the Rankin–Selberg L-function \(L(s,\pi \times \pi ')\). Via the Langlands–Shahidi method, this extends meromorphically to the entire complex plane with at most a simple pole at \(s = 1\), with this pole occurring precisely when \(\pi ' \cong \widetilde{\pi }\). Moreover, this method shows that \(L(s,\pi \times \pi ')\) is nonvanishing in the closed right half-plane \(\mathfrak {R}(s) \ge 1\) [25, Theorem].

Remark 1.2

One can also obtain the nonvanishing of \(L(s,\pi \times \pi ')\) on the line \(\mathfrak {R}(s) = 1\) via the Rankin–Selberg method. For \(n = n'\) and \(\pi ' \ncong \widetilde{\pi }\), this is shown in [21, Theorem 6.1]; the method of proof nonetheless is equally valid for \(n \ne n'\) or \(\pi ' \cong \widetilde{\pi }\), noting in the latter case that \(L(s,\pi \times \widetilde{\pi })\) has a simple pole at \(s = 1\) (see also [24, Equation (1.5)]). Note, however, that the product of L-functions considered in [21, Remark, p. 198] may not be used to show the desired nonvanishing of \(L(1 + it,\pi \times \pi ')\), but merely the nonvanishing of \(L(1,\pi \times \pi ')\).

Proving zero-free regions for \(L(s,\pi \times \widetilde{\pi })\) akin to (1.1), on the other hand, seems to be much more challenging. The method of de la Valleé–Poussin relies on the fact that the Rankin–Selberg convolutions \(L(s,\pi \times \pi )\) and \(L(s,\pi \times \widetilde{\pi })\) exist and extend meromorphically to \(\mathbb {C}\) with at most a simple pole at \(s = 1\). For \(L(s,\pi \times \pi ')\), the associated Rankin–Selberg convolutions have yet to be proved to have these properties, so as yet this method is inapplicable.

Remark 1.3

Note that in [12, Exercise 4, p. 108], it is claimed that one can use this method to prove a zero-free region similar to (1.1) when \(\pi ' \ncong \pi \) and \(\pi ' \ncong \widetilde{\pi }\); however, the hint to this exercise is invalid, as the Dirichlet coefficients of the logarithmic derivative of the auxiliary L-function suggested in this hint are real but not necessarily nonpositive. (In particular, as stated, [12, Exercise 4, p. 108] would imply the nonexistence of Landau–Siegel zeroes upon taking f to be a quadratic Dirichlet character and g to be the trivial character.)

Remark 1.4

When at least one of \(\pi \) and \(\pi '\) is self-dual, then this method can be used to prove a zero-free region akin to (1.1). When both \(\pi \) and \(\pi '\) are self-dual, this is proved by Moreno [21, Theorem 3.3] (see also [24, Equation (1.6)]). When only one of \(\pi \) and \(\pi '\) is self-dual, such a zero-free region has been stated by various authors (in particular, see [5, p. 619], [6, p. 92], and [7, p. 1]); to the best of our knowledge, however, no proof of this claim has appeared in the literature. In the appendix to this article written by Farrell Brumley, a complete proof of this result is given.

In [5], Gelbart and Lapid generalise Sarnak’s effectivization of the Langlands–Shahidi method for \(\zeta (s)\) [24] to prove a zero-free region for \(L(s,\pi \times \pi ')\) of the form
$$\begin{aligned} \sigma \ge 1 - \frac{c_{\pi ,\pi '}}{|t|^{N_{\pi ,\pi '}}} \end{aligned}$$
for some positive constants \(c_{\pi ,\pi '}, N_{\pi ,\pi '}\) dependent on \(\pi \) and \(\pi '\), provided that |t| is sufficiently large; their method applies not only to automorphic representations of \(\mathrm {GL}_n(\mathbb {A}_F)\) but to more general reductive groups.
In [3] and [18, Appendix], Brumley proves a more explicit zero-free region for \(L(s,\pi \times \pi ')\) that is also valid in the analytic conductor aspect and not just the t-aspect. For \(\pi ' \ncong \widetilde{\pi }\), this is of the form
$$\begin{aligned} \sigma \ge 1 - c \left( \left( \mathfrak {q}(\pi ) \mathfrak {q}(\pi ')\right) ^{2(n + n')} (|t| + 3)^{2nn' [F : \mathbb {Q}]}\right) ^{-\frac{1}{2} + \frac{1}{2(n + n')} - \varepsilon }, \end{aligned}$$
together with the bound
$$\begin{aligned} L(s,\pi \times \pi ') \gg _{\varepsilon } \left( \left( \mathfrak {q}(\pi ) \mathfrak {q}(\pi ')\right) ^{2(n + n')} (|t| + 3)^{2nn' [F : \mathbb {Q}]}\right) ^{-\frac{1}{2} + \frac{1}{2(n + n')} - \varepsilon } \end{aligned}$$
for s in this zero-free region, while for \(\pi ' \cong \widetilde{\pi }\), this is of the form
$$\begin{aligned} \sigma \ge 1 - c\left( \mathfrak {q}(\pi )^{8n} (|t| + 3)^{2n^2 [F : \mathbb {Q}]}\right) ^{-\frac{7}{8} + \frac{5}{8n} - \varepsilon }, \end{aligned}$$
(1.5)
together with the bound
$$\begin{aligned} L(s,\pi \times \widetilde{\pi }) \gg _{\varepsilon } \left( \mathfrak {q}(\pi )^{8n} (|t| + 3)^{2n^2 [F : \mathbb {Q}]}\right) ^{-\frac{7}{8} + \frac{5}{8n} - \varepsilon } \end{aligned}$$
(1.6)
for s in this zero-free region.
Recently, Goldfeld and Li [7] have given a strengthening in the t-aspect of a particular case of Brumley’s result, namely the case \(\pi ' \cong \widetilde{\pi }\) subject to the restriction that \(F = \mathbb {Q}\) and that \(\pi \) is unramified and tempered at every nonarchimedean place outside a set of Dirichlet density zero. With these assumptions, they prove the lower bound
$$\begin{aligned} L(1 + it,\pi \times \widetilde{\pi }) \gg _{\pi } \frac{1}{(\log (|t| + 3))^3} \end{aligned}$$
(1.7)
for \(|t| \ge 1\), which gives a zero-free region of the form
$$\begin{aligned} \sigma \ge 1 - \frac{c_{\pi }}{(\log (|t| + 3))^5} \end{aligned}$$
(1.8)
for some positive constant \(c_{\pi }\) dependent on \(\pi \) provided that \(|t| \ge 1\). Their proof, like that of Gelbart and Lapid [5], makes use of Sarnak’s effectivization of the Langlands–Shahidi method; the chief difference is that, like Sarnak but unlike Gelbart and Lapid, they are able to use sieve theory to obtain a much stronger zero-free region. On the downside, the proof is extremely long and technical, and, being written in the classical language instead of the adèlic language, any generalisation of their method to arbitrary number fields and allowing ramification of \(\pi \) would be a challenging endeavour. (Indeed, the Langlands–Shahidi method, in practice, is rather inexplicit at ramified places, though see [11] for explicit calculations for the case \(n = 1\) and \(F = \mathbb {Q}\), so that \(\pi \) corresponds to a primitive Dirichlet character.)

In this article, we give a simple proof of the following.

Theorem 1.9

Let \(\pi \) be a unitary cuspidal automorphic representation of \(\mathrm {GL}_n(\mathbb {A}_F)\) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero. Then there exists an absolute constant \(c_{\pi }\) dependent on \(\pi \) (and hence also on n and F) such that \(L(s,\pi \times \widetilde{\pi })\) has no zeroes in the region
$$\begin{aligned} \sigma \ge 1 - \frac{c_{\pi }}{\log (|t| + 3)} \end{aligned}$$
(1.10)
with \(|t| \ge 1\). Furthermore, we have the bound
$$\begin{aligned} L(s,\pi \times \widetilde{\pi }) \gg _{\pi } \frac{1}{\log (|t| + 3)} \end{aligned}$$
(1.11)
for s in this region.

In particular, we improve the zero-free region (1.8) and lower bound (1.7) of Goldfeld and Li to (1.10) and (1.11) respectively while removing Goldfeld and Li’s restriction that \(F = \mathbb {Q}\) and that \(\pi \) is unramified at every place. Nonetheless, we still require that \(\pi \) be tempered at every nonarchimedean place outside a set of Dirichlet density zero; moreover, this zero-free region is only in the t-aspect, unlike Brumley’s zero-free region in the analytic conductor aspect.

The proof of Theorem 1.9 shares some similarities with the method of de la Valleé–Poussin. Once again, one creates an auxiliary L-function, though this has a zero of equal order to the order of the pole at \(s = 1\). While Landau’s lemma cannot be used in this setting to obtain a standard zero-free region, one can instead use sieve theory. This approach is discussed in [26, Section 3.8] when \(L(s,\pi \times \widetilde{\pi })\) is the Riemann zeta function, so that \(F = \mathbb {Q}\) and \(\pi \) is trivial, and this method can also be adapted to prove a standard zero-free region in the q-aspect for \(L(s,\chi )\), where \(\chi \) is a primitive Dirichlet character; cf. [1, 11].

This usage of sieve theory, however, seems to have limitations; it is unclear how to prove zero-free regions via sieve theory for Rankin–Selberg L-functions \(L(s,\pi \times \pi ')\) with \(\pi ' \ncong \widetilde{\pi }\), even in the particular case \(\pi ' = 1\) and \(n \ge 2\), so that \(L(s,\pi \times \pi ')\) is the standard L-function \(L(s,\pi )\). We expand upon this point in Remark 3.5.

By slightly different means, we sketch how to prove a weaker version of Theorem 1.9.

Theorem 1.12

Let \(\pi \) be a unitary cuspidal automorphic representation of \(\mathrm {GL}_n(\mathbb {A}_F)\) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero. Then for \(|t| \ge 1\), we have the bound
$$\begin{aligned} L(1 + it,\pi \times \widetilde{\pi }) \gg _{\pi } \frac{1}{(\log (|t| + 3))^3}, \end{aligned}$$
(1.13)
and so there exists an absolute constant \(c_{\pi }\) dependent on \(\pi \) such that \(L(s,\pi \times \widetilde{\pi })\) has no zeroes in the region
$$\begin{aligned} \sigma \ge 1 - \frac{c_{\pi }}{(\log (|t| + 3))^5}. \end{aligned}$$
(1.14)

Though this is a weaker result than Theorem 1.9, the method of proof is of particular interest; it is essentially a generalisation from \(\mathrm {GL}_1(\mathbb {A}_{\mathbb {Q}})\) to \(\mathrm {GL}_n(\mathbb {A}_F)\) of the method of Balasubramanian and Ramachandra [1]. It turns out that Brumley’s method [3] in proving (1.6) is a natural generalisation of [1] except that sieve theory is not used and so the resulting lower bounds for \(L(1 + it,\pi \times \widetilde{\pi })\) are not nearly as strong.

Theorem 1.12 gives the same bounds as obtained by Goldfeld and Li, and this is no accident. Goldfeld and Li create an integral of an Eisenstein series and obtain upper bounds for this integral via the Maaß–Selberg relation together with upper bounds for \(L(1 + it, \pi \times \widetilde{\pi })\) and \(L'(1 + it, \pi \times \widetilde{\pi })\), while they use the Fourier expansion of the Eisenstein series together with sieve theory to find lower bounds for this integral. In the proof of Theorem 1.12, we follow Brumley’s method of studying a smoothed average of the Dirichlet coefficients of an auxiliary L-function. Upper bounds for this smoothed average are then obtained via Perron’s inversion formula and Cauchy’s residue theorem, in place of Goldfeld and Li’s usage of the Maaß–Selberg relation, together with upper bounds for \(L(1 + it, \pi \times \widetilde{\pi })\) and \(L'(1 + it, \pi \times \widetilde{\pi })\); lower bounds for this smoothed average stem once again from sieve theory.

2 Sieve theory

The L-function \(L(s,\pi )\) of \(\pi \) can be written as the Dirichlet series
$$\begin{aligned} L(s,\pi ) = \sum _{\begin{array}{c} \mathfrak {a}\subset \mathcal {O}_F \\ \mathfrak {a}\ne \{0\} \end{array}} \frac{\lambda _{\pi }(\mathfrak {a})}{N(\mathfrak {a})^s} \end{aligned}$$
for \(\mathfrak {R}(s)\) sufficiently large, where \({N(\mathfrak {a}) = N_{F/\mathbb {Q}}(\mathfrak {a}) :=\# \mathcal {O}_F / \mathfrak {a}}\), and extends to a meromorphic function on \(\mathbb {C}\) with at most a simple pole at \(s = 1\) if \(n = 1\) and \(\pi \) is trivial, so that \(L(s,\pi ) = \zeta _F(s)\). Similarly, the Rankin–Selberg L-function \(L(s,\pi \times \widetilde{\pi })\) is meromorphic on \(\mathbb {C}\) with only a simple pole at \(s = 1\). We denote by \(\Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {a})\) the coefficients of the Dirichlet series for \(-\frac{L'}{L}(s,\pi \times \widetilde{\pi })\), so that
$$\begin{aligned} -\frac{L'}{L}(s,\pi \times \widetilde{\pi }) = \sum _{\begin{array}{c} \mathfrak {a}\subset \mathcal {O}_F \\ \mathfrak {a}\ne \{0\} \end{array}} \frac{\Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {a})}{N(\mathfrak {a})^s}. \end{aligned}$$
These coefficients are nonnegative; see [12, Remark, p. 138]. Moreover, the residue of this at \(s = 1\) is 1, and we have that
$$\begin{aligned} \Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {p}) = \left| \lambda _{\pi }(\mathfrak {p})\right| ^2 \log N(\mathfrak {p}) \end{aligned}$$
whenever \(\pi \) is unramified at \(\mathfrak {p}\).

We denote by \(S_{\pi }\) the set of places of F at which \(\pi \) is either ramified or nontempered.

Lemma 2.1

([7, Lemmata 12.12 and 12.15]) Suppose that \(\pi \) is tempered at every nonarchimedean place outside a set of Dirichlet density zero. For \(Y \gg _{\pi } (|t| + 3)^2\),
$$\begin{aligned} \sum _{\begin{array}{c} Y \le N(\mathfrak {p}) \le 2Y \\ \mathfrak {p}\notin S_{\pi } \end{array}} \left| \lambda _{\pi }(\mathfrak {p}) \right| ^2 \left| 1 + N(\mathfrak {p})^{it}\right| ^2 \gg _{\pi } \frac{Y}{\log Y}. \end{aligned}$$

Proof

We use Ikehara’s Tauberian theorem and the fact that \(S_{\pi }\) has Dirichlet density zero to see that
$$\begin{aligned} \sum _{\begin{array}{c} Y \le N(\mathfrak {p}) \le 2Y \\ \mathfrak {p}\notin S_{\pi } \end{array}} \left| \lambda _{\pi }(\mathfrak {p})\right| ^2 \log N(\mathfrak {p}) = \sum _{Y \le N(\mathfrak {a}) \le 2Y} \Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {a}) + o_{\pi }(Y) = Y + o_{\pi }(Y). \end{aligned}$$
(2.2)
The assumption that \(\pi \) is tempered at every nonarchimedean place outside a set of Dirichlet density zero implies that \(|\lambda _{\pi }(\mathfrak {p})| \le n\) whenever \(\mathfrak {p}\notin S_{\pi }\), so that for any \(C > 0\), the left-hand side of (2.2) is
$$\begin{aligned}&\sum _{\begin{array}{c} Y \le N(\mathfrak {p}) \le 2Y \\ \mathfrak {p}\notin S_{\pi } \\ |\lambda _{\pi }(\mathfrak {p})| < C \end{array}} \left| \lambda _{\pi }(\mathfrak {p})\right| ^2 \log N(\mathfrak {p}) + \sum _{\begin{array}{c} Y \le N(\mathfrak {p}) \le 2Y \\ \mathfrak {p}\notin S_{\pi } \\ |\lambda _{\pi }(\mathfrak {p})| \ge C \end{array}} \left| \lambda _{\pi }(\mathfrak {p})\right| ^2 \log N(\mathfrak {p}) \\&\quad \le C^2 \sum _{Y \le N(\mathfrak {p}) \le 2Y} \log N(\mathfrak {p}) + n^2 \log 2Y \#\left\{ Y \le N(\mathfrak {p}) \le 2Y : \mathfrak {p}\notin S_{\pi }, \ |\lambda _{\pi }(\mathfrak {p})| \ge C \right\} , \end{aligned}$$
and as
$$\begin{aligned} \sum _{Y \le N(\mathfrak {p}) \le 2Y} \log N(\mathfrak {p}) = \sum _{Y \le N(\mathfrak {a}) \le 2Y} \Lambda (\mathfrak {a}) + o_F(Y) = Y + o_F(Y), \end{aligned}$$
we ascertain that
$$\begin{aligned} \#\left\{ Y \le N(\mathfrak {p}) \le 2Y : \mathfrak {p}\notin S_{\pi }, \ |\lambda _{\pi }(\mathfrak {p})| \ge C \right\} \ge \frac{1 - C^2}{n^2} \frac{Y}{\log Y} + o_{\pi }\left( \frac{Y}{\log Y}\right) . \end{aligned}$$
(2.3)
Next, for \(C \in (0,2)\), we note that
$$\begin{aligned} \left| 1 + N(\mathfrak {p})^{it}\right| = 2\left| \sin \left( \frac{|t|}{2} \log N(\mathfrak {p}) - (2m - 1) \frac{\pi }{2}\right) \right| \end{aligned}$$
for any integer m, and so via the bound \(|\sin x| \le |x|\), we have that
$$\begin{aligned}&\#\left\{ Y \le N(\mathfrak {p}) \le 2Y : \left| 1 + N(\mathfrak {p})^{it}\right| < C \right\} \nonumber \\&\quad \le \sum _{\frac{|t|}{2\pi } \log Y - \frac{C}{2\pi } + \frac{1}{2} \le m \le \frac{|t|}{2\pi } \log 2Y + \frac{C}{2\pi } + \frac{1}{2}} \#\left\{ e^{\frac{(2m - 1)\pi - C}{|t|}} \le N(\mathfrak {p}) \le e^{\frac{(2m - 1)\pi + C}{|t|}} \right\} . \end{aligned}$$
(2.4)
From [9, Proposition 2], we have that
$$\begin{aligned} \pi _F(x + y) - \pi _F(x) \le 4 [F : \mathbb {Q}] \frac{y}{\log y} \end{aligned}$$
for \(2 \le y \le x\), where \({\pi _F(x) :=\#\{N(\mathfrak {p}) \le x\}}\); the proof of this reduces to the case \(F = \mathbb {Q}\), in which case this is a well-known result that can be proven via the Selberg sieve (with the appearance of an additional error term) or the large sieve. So assuming that \(\frac{1}{2\sqrt{Y}} \le C \le \frac{|t| \log 2}{2}\) and \(Y > 4|t|^2\), the inner term on the right-hand side of (2.4) is bounded by
$$\begin{aligned} 64 [F : \mathbb {Q}] \frac{CY}{|t| \log \frac{Y}{4|t|^2}} \end{aligned}$$
using the fact that \(\log (e^u + 1) \ge \log u\) and \(e^u - 1 \le 2u\) for \(u \in (0,1)\). Consequently,
$$\begin{aligned} \#\left\{ Y \le N(\mathfrak {p}) \le 2Y : \left| 1 + N(\mathfrak {p})^{it}\right| < C \right\} \le \frac{64 C [F : \mathbb {Q}] \log 2}{\pi } \frac{Y}{\log \frac{Y}{4|t|^2}}. \end{aligned}$$
Since
$$\begin{aligned} \#\left\{ Y \le N(\mathfrak {p}) \le 2Y : Y \notin S_{\pi }\right\} = \frac{Y}{\log Y} + o_F\left( \frac{Y}{\log Y}\right) , \end{aligned}$$
it follows that for \(Y \gg _F (|t| + 3)^2\),
$$\begin{aligned}&\#\left\{ Y \le N(\mathfrak {p}) \le 2Y : \mathfrak {p}\notin S_{\pi }, \ \left| 1 + N(\mathfrak {p})^{it}\right| \ge C \right\} \nonumber \\&\quad \ge \left( 1 - \frac{64 C [F : \mathbb {Q}] \log 2}{\pi }\right) \frac{Y}{\log Y} + o_{\pi }\left( \frac{Y}{\log Y}\right) . \end{aligned}$$
(2.5)
By choosing C sufficiently small in terms of n and F, (2.3) and (2.5) imply that
$$\begin{aligned} \#\left\{ Y \le N(\mathfrak {p}) \le 2Y : \mathfrak {p}\notin S_{\pi }, \ \left| \lambda _{\pi }(\mathfrak {p})\right| \left| 1 + N(\mathfrak {p})^{it}\right| \ge C^2 \right\} \gg _{\pi ,C} \frac{Y}{\log Y}, \end{aligned}$$
from which the result follows. \(\square \)

Remark 2.6

The only point at which we make use of the assumption that \(\pi \) is tempered at every nonarchimedean place outside a set of Dirichlet density zero is in proving (2.3). It would be of interest whether an estimate akin to (2.3) could be proved unconditionally.

Remark 2.7

While the implicit constants in Theorems 1.9 and 1.12 depend on \(\pi \), much of the argument still works if we keep track of this dependence in terms of the analytic conductor of \(\pi \). The main issue seems to be the lower bound stemming from Lemma 2.1; in particular, the use of Ikehara’s Tauberian theorem to prove (2.2). We could instead use (1.6) together with an upper bound for \(L'(\sigma + it, \pi \times \widetilde{\pi })\) in the region (1.5) derived via the methods of Li [19] to prove (2.2) with an error term that is effective in terms of the analytic conductor of \(\pi \), but the payoff would not be great as the weaker zero-free region (1.5) would only give a weak error term.

3 Proof of Theorem 1.9

Let \(\pi \) and \(\pi '\) be unitary cuspidal automorphic representations of \(\mathrm {GL}_n(\mathbb {A}_F)\) and \(\mathrm {GL}_{n'}(\mathbb {A}_F)\) respectively. Let \(\rho = \beta + i\gamma \) be a nontrivial zero of \(L(s,\pi \times \pi ')\) with \(1/2 \le \beta < 1\) and \(\gamma \ne 0\). We define
$$\begin{aligned} {\Pi :=\pi \otimes \left| \det \right| ^{\frac{i\gamma }{2}} \boxplus \widetilde{\pi }' \otimes \left| \det \right| ^{-\frac{i\gamma }{2}},} \end{aligned}$$
This is an isobaric (noncuspidal) automorphic representation of \(\mathrm {GL}_{n + n'}(\mathbb {A}_F)\). The Rankin–Selberg L-function of \(\Pi \) and \(\widetilde{\Pi }\) factorises as
$$\begin{aligned} L(s,\Pi \times \widetilde{\Pi }) = L(s, \pi \times \widetilde{\pi }) L(s, \pi ' \times \widetilde{\pi }') L(s + i\gamma , \pi \times \pi ') L(s - i\gamma , \widetilde{\pi } \times \widetilde{\pi }'). \end{aligned}$$
(3.1)
This is a meromorphic function on \(\mathbb {C}\) with a double pole at \(s = 1\), simple poles at \(s = 1 \pm i\gamma \) if \(\pi ' \cong \widetilde{\pi }\), and holomorphic elsewhere. We let \(\Lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a})\) denote the coefficients of the Dirichlet series for \(-\frac{L'}{L}(s,\Pi \times \widetilde{\Pi })\), so that
$$\begin{aligned} -\frac{L'}{L}(s,\Pi \times \widetilde{\Pi }) = \sum _{\begin{array}{c} \mathfrak {a}\subset \mathcal {O}_F \\ \mathfrak {a}\ne \{0\} \end{array}} \frac{\Lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a})}{N(\mathfrak {a})^s}. \end{aligned}$$
Again, these coefficients are nonnegative.

Lemma 3.2

For \(\sigma > 1\),
$$\begin{aligned} -\frac{L'}{L}(\sigma ,\Pi \times \widetilde{\Pi }) < -\frac{2}{\sigma - \beta } + \frac{2}{\sigma - 1} + O\left( \log \mathfrak {q}(\Pi \times \widetilde{\Pi })\right) . \end{aligned}$$

Proof

By taking the real part of [12, (5.28)], we have that
$$\begin{aligned} -\frac{L'}{L}(\sigma + i\gamma ,\pi \times \pi ') - \frac{L'}{L}(\sigma - i\gamma ,\widetilde{\pi } \times \widetilde{\pi }') < -\frac{2}{\sigma - \beta } + O\left( \log \mathfrak {q}(i\gamma ,\pi \times \pi ')\right) \end{aligned}$$
for \(\sigma > 1\); cf. [12, (5.37)]. Similarly,
$$\begin{aligned} -\frac{L'}{L}(\sigma ,\pi \times \widetilde{\pi }) - \frac{L'}{L}(\sigma ,\pi ' \times \widetilde{\pi }') < \frac{2}{\sigma - 1} + O\left( \log \mathfrak {q}(\pi \times \widetilde{\pi }) \mathfrak {q}(\pi ' \times \widetilde{\pi }')\right) \end{aligned}$$
for \(\sigma > 1\) via [12, (5.37)], using the fact that \(\Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {a})\) and \(\Lambda _{\pi ' \times \widetilde{\pi }'}(\mathfrak {a})\) are real. \(\square \)

Lemma 3.3

Suppose that \(\pi \) and \(\pi '\) are unramified at \(\mathfrak {p}\). Then
$$\begin{aligned} \Lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {p}) = \log N(\mathfrak {p}) \left| \lambda _{\pi }(\mathfrak {p}) + \lambda _{\widetilde{\pi }'}(\mathfrak {p}) N(\mathfrak {p})^{i\gamma }\right| ^2. \end{aligned}$$

Proof

Indeed, (3.1) implies that
$$\begin{aligned} \Lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {p}) = \Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {p}) + \Lambda _{\pi ' \times \widetilde{\pi }'}(\mathfrak {p}) + \Lambda _{\pi \times \pi '}(\mathfrak {p}) N(\mathfrak {p})^{-i\gamma } + \Lambda _{\widetilde{\pi } \times \widetilde{\pi }'}(\mathfrak {p}) N(\mathfrak {p})^{i\gamma }, \end{aligned}$$
and
$$\begin{aligned} \Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {p})= & {} \log N(\mathfrak {p}) \left| \lambda _{\pi }(\mathfrak {p})\right| ^2, \\ \Lambda _{\pi ' \times \widetilde{\pi }'}(\mathfrak {p})= & {} \log N(\mathfrak {p}) \left| \lambda _{\pi '}(\mathfrak {p})\right| ^2, \\ \Lambda _{\pi \times \pi '}(\mathfrak {p})= & {} \log N(\mathfrak {p}) \lambda _{\pi }(\mathfrak {p}) \lambda _{\pi '}(\mathfrak {p}), \\ \Lambda _{\widetilde{\pi } \times \widetilde{\pi }'}(\mathfrak {p})= & {} \log N(\mathfrak {p}) \overline{\lambda _{\pi }(\mathfrak {p}) \lambda _{\pi '}(\mathfrak {p})} \end{aligned}$$
whenever \(\pi \) and \(\pi '\) are unramified at \(\mathfrak {p}\). \(\square \)

Now let us restrict to the case \(\pi ' = \widetilde{\pi }\).

Corollary 3.4

Suppose that \(\pi \) is tempered at every nonarchimedean place outside a set of Dirichlet density zero. Then for \(\sigma > 1\),
$$\begin{aligned} -\frac{L'}{L}(\sigma ,\Pi \times \widetilde{\Pi }) \gg _{\pi } \frac{(|\gamma | + 3)^{2(1 - \sigma )}}{\sigma - 1}. \end{aligned}$$

Proof

We have that
$$\begin{aligned} -\frac{L'}{L}(\sigma ,\Pi \times \widetilde{\Pi })&\ge \sum _{\begin{array}{c} N(\mathfrak {p}) \gg _{\pi } (|\gamma | + 3)^2 \\ \mathfrak {p}\notin S_{\pi } \end{array}} \frac{\log N(\mathfrak {p})}{N(\mathfrak {p})^{\sigma }} \left| \lambda _{\pi }(\mathfrak {p})\right| ^2 \left| 1 + N(\mathfrak {p})^{i\gamma }\right| ^2 \\&\gg _{\pi } \frac{(|\gamma | + 3)^{2(1 - \sigma )}}{\sigma - 1} \end{aligned}$$
by dividing into dyadic intervals and applying Lemma 2.1.

Proof of Theorem 1.9

By combining Lemma 3.2 and Corollary 3.4 and choosing \(\sigma = 1 + c/\log (|\gamma | + 3)\), we find that
$$\begin{aligned} 1 - \beta \gg _{\pi } \frac{1}{\log (|\gamma | + 3)}, \end{aligned}$$
which gives the zero-free region (1.10). Now using [12, (5.28)], we find in the region
$$\begin{aligned} \sigma \ge 1 - \frac{c_{\pi }}{2\log (|t| + 3)} \end{aligned}$$
away from \(t = 0\), we have that
$$\begin{aligned} -\frac{L'}{L}(s,\pi \times \widetilde{\pi }) \ll _{\pi } \log (|t| + 3). \end{aligned}$$
Next, we note that
$$\begin{aligned} \log L(s, \pi \times \widetilde{\pi }) = \sum _{\begin{array}{c} \mathfrak {a}\subset \mathcal {O}_F \\ \mathfrak {a}\notin \{\{0\},\mathcal {O}_F\} \end{array}} \frac{\Lambda _{\pi \times \widetilde{\pi }}(\mathfrak {a})}{N(\mathfrak {a})^s \log N(\mathfrak {a})} \end{aligned}$$
for \(\mathfrak {R}(s) > 1\). So
$$\begin{aligned} \left| \log L(s, \pi \times \widetilde{\pi })\right| \le \log L(\sigma , \pi \times \widetilde{\pi }). \end{aligned}$$
Since \(L(s,\pi \times \widetilde{\pi })\) has a simple pole at \(s = 1\),
$$\begin{aligned} \log L(s, \pi \times \widetilde{\pi }) \ll _{\pi } \log \frac{1}{\sigma - 1}. \end{aligned}$$
In particular, in the region
$$\begin{aligned} \sigma \ge 1 + \frac{1}{\log (|t| + 3)}, \end{aligned}$$
we have that
$$\begin{aligned} \log L(s, \pi \times \widetilde{\pi }) \ll _{\pi } \log \log (|t| + 3). \end{aligned}$$
Now suppose that \(s = \sigma + it\) with
$$\begin{aligned} 1 - \frac{c_{\pi }}{2\log (|t| + 3)} \le \sigma \le 1 + \frac{1}{\log (|t| + 3)}. \end{aligned}$$
Then \(\log L(s, \pi \times \widetilde{\pi })\) is equal to
$$\begin{aligned} \log L\left( 1 + \frac{1}{\log (|t| + 3)} + it, \pi \times \widetilde{\pi }\right) + \int _{1 + \frac{1}{\log (|t| + 3)} + it}^{s} \frac{L'}{L}(w,\pi \times \widetilde{\pi }) \, dw, \end{aligned}$$
so again
$$\begin{aligned} \log L(s, \pi \times \widetilde{\pi }) \ll _{\pi } \log \log (|t| + 3). \end{aligned}$$
Finally, we note that
$$\begin{aligned}\frac{1}{\left| L(s,\pi \times \widetilde{\pi })\right| } = \exp \left( -\mathfrak {R}\left( \log L(s,\pi \times \widetilde{\pi })\right) \right) \ll _{\pi } \log (|t| + 3), \end{aligned}$$
which is equivalent to (1.11). \(\square \)

Remark 3.5

To prove Theorem 1.9 for \(L(s,\pi \times \pi ')\) with \(\pi ' \ncong \widetilde{\pi }\), we would need to replace Lemma 2.1 with a result of the form
$$\begin{aligned}\sum _{\begin{array}{c} Y \le N(\mathfrak {p}) \le 2Y \\ \mathfrak {p}\notin S_{\pi } \cup S_{\pi '} \end{array}} \left| \lambda _{\pi }(\mathfrak {p}) + \lambda _{\widetilde{\pi }'}(\mathfrak {p}) N(\mathfrak {p})^{i\gamma }\right| ^2 \gg _{\pi ,\pi '} \frac{Y}{\log Y}, \end{aligned}$$
but it is unclear how one might generalise the proof of Lemma 2.1 to obtain such a result.

4 Proof of Theorem 1.12

For \(t \in \mathbb {R}{\setminus } \{0\}\), define the isobaric automorphic representation \(\Pi \) of \(\mathrm {GL}_{n + n'}(\mathbb {A}_F)\) by
$$\begin{aligned} {\Pi :=\pi \otimes \left| \det \right| ^{\frac{it}{2}} \boxplus \widetilde{\pi }' \otimes \left| \det \right| ^{-\frac{it}{2}}.} \end{aligned}$$
Then
$$\begin{aligned} L(s,\Pi \times \widetilde{\Pi }) = L(s, \pi \times \widetilde{\pi }) L(s, \pi ' \times \widetilde{\pi }') L(s + it, \pi \times \pi ') L(s - it, \widetilde{\pi } \times \widetilde{\pi }'). \end{aligned}$$
(4.1)
This is a meromorphic function on \(\mathbb {C}\) with a double pole at \(s = 1\), simple poles at \(s = 1 \pm it\) if \(\pi ' \cong \widetilde{\pi }\), and holomorphic elsewhere.
We let \(\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a})\) denote the coefficients of the Dirichlet series for \(L(s,\Pi \times \widetilde{\Pi })\), so that
$$\begin{aligned} L(s,\Pi \times \widetilde{\Pi }) = \sum _{\begin{array}{c} \mathfrak {a}\subset \mathcal {O}_F \\ \mathfrak {a}\ne \{0\} \end{array}} \frac{\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a})}{N(\mathfrak {a})^s}. \end{aligned}$$
Again, the coefficients \(\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a})\) are nonnegative. Write
$$\begin{aligned} L(s,\Pi \times \widetilde{\Pi }) = \frac{r_{-2}}{(s - 1)^2} + \frac{r_{-1}}{s - 1} + O(1) \end{aligned}$$
for s near 1 and
$$\begin{aligned} L(s,\Pi \times \widetilde{\Pi }) = \delta _{\pi ',\widetilde{\pi }} \frac{r_{-1}^{\pm }}{s - (1 \pm it)} + O(1) \end{aligned}$$
for s near \(1 \pm it\). Finally, we write
$$\begin{aligned} \zeta _F(s) = \frac{\gamma _{-1}(F)}{s - 1} + \gamma _0(F) + O(s - 1) \end{aligned}$$
for s near 1.

Lemma 4.2

We have that
$$\begin{aligned} \frac{r_{-2}}{\left| L(1 + it, \pi \times \pi ')\right| ^2}= & {} \gamma _{-1}(F)^2 L(1, {{\mathrm{ad}}}\pi ) L(1, {{\mathrm{ad}}}\pi '), \\ \frac{r_{-1}}{\left| L(1 + it, \pi \times \pi ')\right| ^2}= & {} L(1, {{\mathrm{ad}}}\pi ) (\gamma _0(F) + \gamma _{-1}(F) L'(1, {{\mathrm{ad}}}\pi ')) \\&+\, L(1, {{\mathrm{ad}}}\pi ') (\gamma _0(F) + \gamma _{-1}(F) L'(1, {{\mathrm{ad}}}\pi )) \\&+\, 2 \gamma _{-1}(F)^2 L(1, {{\mathrm{ad}}}\pi ) L(1, {{\mathrm{ad}}}\pi ') \mathfrak {R}\left( \frac{L}{L}'(1 + it, \pi \times \pi ')\right) . \end{aligned}$$
Similarly,
$$\begin{aligned} r_{-1}^{\pm } = \gamma _{-1}(F) L(1, {{\mathrm{ad}}}\pi ) L(1 \pm it, \pi \times \widetilde{\pi })^2 L(1 \pm 2it, \pi \times \widetilde{\pi }). \end{aligned}$$

Proof

This follows from the factorisation \(L(s,\pi \times \widetilde{\pi }) = \zeta _F(s) L(s, {{\mathrm{ad}}}\pi )\). \(\square \)

Lemma 4.3

([7, Lemma 5.1]) We have that
$$\begin{aligned} L(1 + it, \pi \times \pi ')&\ll _{\pi ,\pi '} \log (|t| + 3), \\ L'(1 + it, \pi \times \pi ')&\ll _{\pi ,\pi '} (\log (|t| + 3))^2. \end{aligned}$$

Proof

This is proved by Goldfeld and Li in [7, Lemma 5.1] for \(F = \mathbb {Q}\) and \(\pi ' \cong \widetilde{\pi }\), but the proof in this more generalised setting (via the approximate functional equation) follows mutatis mutandis. \(\square \)

Together with the fact that \(L(1,{{\mathrm{ad}}}\pi ) \ne 0\), this shows the following.

Corollary 4.4

We have that
$$\begin{aligned} r_{-2}&\ll _{\pi ,\pi '} \left| L(1 + it, \pi \times \pi ')\right| \log (|t| + 3), \\ r_{-1}&\ll _{\pi ,\pi '} \left| L(1 + it, \pi \times \pi ')\right| (\log (|t| + 3))^2, \\ r_{-1}^{\pm }&\ll _{\pi ,\pi '} \left| L(1 + it, \pi \times \pi ')\right| (\log (|t| + 3))^2. \end{aligned}$$
Now let \(\psi \in C^{\infty }_c(0,\infty )\) be a fixed nonnegative function satisfying \(\psi (x) = 1\) for \(x \in [1,2]\) and \(\psi (0) = 0\). The Mellin transform of \(\psi \) is
$$\begin{aligned} {\widehat{\psi }(s) :=\int _{0}^{\infty } \psi (x) x^s \, \frac{dx}{x},} \end{aligned}$$
which is entire with rapid decay in vertical strips. Define
$$\begin{aligned} F(Y) = \sum _{\begin{array}{c} \mathfrak {a}\subset \mathcal {O}_F \\ \mathfrak {a}\ne \{0\} \end{array}} \lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a}) \psi \left( \frac{N(\mathfrak {a})}{Y}\right) . \end{aligned}$$
We let \(\mathfrak {q}(\Pi \times \widetilde{\Pi })\) denotes the analytic conductor of \(\Pi \times \widetilde{\Pi }\); from [12, (5.11)], we have that
$$\begin{aligned} \mathfrak {q}(\Pi \times \widetilde{\Pi }) \le \left( \mathfrak {q}(\pi ) \mathfrak {q}(\pi ')\right) ^{2(n + n')} (|t| + 3)^{2nn' [F :\mathbb {Q}]}. \end{aligned}$$
(4.5)
On the other hand, it is easily seen that
$$\begin{aligned} \mathfrak {q}(\Pi \times \widetilde{\Pi }) \gg _{\pi ,\pi '} (|t| + 3)^2. \end{aligned}$$

Remark 4.6

While (4.5) is stated in [12, (5.11)], a complete proof does not seem to have appeared in the literature. In the appendix to this article, a proof of (a more general version of) this statement is given.

Lemma 4.7

(Cf. [3, Proof of Theorem 3]) For \(Y \ge \mathfrak {q}(\Pi \times \widetilde{\Pi })\), there exists \(\delta > 0\) such that
$$\begin{aligned} F(Y)= & {} r_{-2} \widehat{\psi }(1) Y \log Y + \left( r_{-1} \widehat{\psi }(1) + r_{-2} \widehat{\psi }'(1)\right) Y \\&+\,\, \delta _{\pi ',\widetilde{\pi }} r_{-1}^{+} \widehat{\psi }(1 + it) Y^{1 + it} + \delta _{\pi ',\widetilde{\pi }} r_{-1}^{-} \widehat{\psi }(1 - it) Y^{1 - it} + O(Y^{1 - \delta }). \end{aligned}$$

Proof

Indeed,
$$\begin{aligned} F(Y) = \int _{\sigma - i\infty }^{\sigma + i\infty } L(s,\Pi \times \widetilde{\Pi }) \widehat{\psi }(s) Y^s \, ds \end{aligned}$$
for \(\sigma > 1\), and moving the contour to the left shows that
$$\begin{aligned} F(Y)= & {} \left( {{\mathrm{Res}}}_{s = 1} + \delta _{\pi ',\widetilde{\pi }} {{\mathrm{Res}}}_{s = 1 + it} + \delta _{\pi ',\widetilde{\pi }} {{\mathrm{Res}}}_{s = 1 - it}\right) L(s,\Pi \times \widetilde{\Pi }) \widehat{\psi }(s) Y^s \\&+\, \frac{1}{2\pi i} \int _{\sigma - i\infty }^{\sigma + i\infty } L(s,\Pi \times \widetilde{\Pi }) \widehat{\psi }(s) Y^s \, ds \end{aligned}$$
for any \(\sigma < 1\). The convexity bound of Li [19] and the rapid decay of \(\widehat{\psi }\) in vertical strips imply that
$$\begin{aligned} \frac{1}{2\pi i} \int _{\sigma - i\infty }^{\sigma + i\infty } L(s,\Pi \times \widetilde{\Pi }) \widehat{\psi }(s) Y^s \, ds \ll _{\varepsilon } \mathfrak {q}(\Pi \times \widetilde{\Pi })^{\frac{1 - \sigma }{2} + \varepsilon } Y^{\sigma }, \end{aligned}$$
from which the result follows. \(\square \)
In [3, 18], Brumley notes that
$$\begin{aligned} F(Y) \ge \sum _{Y \le N(\mathfrak {a}^{2n}) \le 2Y} \lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a}^{2n}) \end{aligned}$$
and that \(\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a}^{2n}) \ge 1\). This is paired with a modified version of Lemma 4.7 in order to prove effective lower bounds for \(\left| L(1 + it, \pi \times \widetilde{\pi })\right| \) in terms of the analytic conductor \(\mathfrak {q}(\pi \times \widetilde{\pi }, 1 + it)\). Instead of restricting the sum over integral ideals to those that are 2n-powers and using the fact that \(\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {a}^{2n}) \ge 1\), our approach is to restrict to prime ideals at which \(\pi \) is unramified and tempered and use sieve theory to show that \(\lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {p})\) is often not too small on dyadic intervals.

Lemma 4.8

Suppose that \(\pi \) and \(\pi '\) are unramified at \(\mathfrak {p}\). Then
$$\begin{aligned} \lambda _{\Pi \times \widetilde{\Pi }}(\mathfrak {p}) = \left| \lambda _{\pi }(\mathfrak {p}) + \lambda _{\widetilde{\pi }'}(\mathfrak {p}) N(\mathfrak {p})^{it}\right| ^2. \end{aligned}$$

Proof

This follows via the same method as the proof of Lemma 3.3. \(\square \)

We now restrict to the case \(\pi ' = \widetilde{\pi }\).

Proof of Theorem 1.12

From Lemma 4.7 and Corollary 4.4, we have that for \(Y \ge \mathfrak {q}(\Pi \times \widetilde{\Pi })\), there exists \(\delta > 0\) such that
$$\begin{aligned} F(Y) \ll _{\pi } \left| L(1 + it, \pi \times \widetilde{\pi })\right| Y (\log Y)^2 + Y^{1 - \delta }. \end{aligned}$$
On the other hand, Lemmata 4.8 and 2.1 imply that for \(Y \gg _{\pi } (|t| + 3)^2\),
$$\begin{aligned} F(Y) \gg _{\pi } \frac{Y}{\log Y}. \end{aligned}$$
This gives the lower bound (1.13). Then as in [7, Proof of Theorem 1.2], (1.14) follows via the mean value theorem. \(\square \)

Footnotes

  1. 1.

    Supported by ANR grant 14-CE25.

  2. 2.

    I would like thank Philippe Michel and Étienne Fouvry for suggesting that I write up a proof of Theorem A.1. I am grateful as well to Peter Humphries for allowing me to include this appendix to his paper, and for suggesting many improvements to the proofs and exposition.

Notes

Acknowledgements

The author would like to thank Peter Sarnak, Dorian Goldfeld, and Farrell Brumley for helpful discussions and comments, as well as Dimitris Koukoulopoulos for alerting the author to an error in an earlier version of this article.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.LAGA-Institut GaliléeVilletaneuseFrance

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