1 Introduction

Let \(s=\sigma +\textrm{i}t\), where \(\sigma \) and t are real numbers. Determination of the true order of \(\left| \zeta (s)\right| \) in the critical strip, or any other respectable L-function, is one of the great problems in zeta-function theory with far-reaching consequences in analytic number theory. It is believed that \(\zeta \left( \sigma +\textrm{i}t\right) \ll _{\varepsilon ,\sigma } |t|^{\varepsilon }\) for every \(\varepsilon >0\) and \(\sigma \ge 1/2\), which in the case \(\sigma =1/2\) is known as the Lindelöf hypothesis, but any unconditional approach to such bounds seems to be a very hard problem, e.g. see [2] for the latest result when \(\sigma =1/2\). For some explicit results in this direction, see [12,13,14, 23, 24, 34, 35].

Assuming the Riemann hypothesis (RH), Littlewood proved in 1912 that

$$\begin{aligned} \log {\zeta (s)} \ll _{\varepsilon ,\sigma _0} \left( \log {t}\right) ^{2(1-\sigma )+\varepsilon } \end{aligned}$$
(1)

for \(\varepsilon >0\), \(1/2<\sigma _0\le \sigma \le 1\) and t large. Techniques in the proof ([33, Theorem 14.2], [15, Theorem 1.12]) are purely complex analytic: the Hadamard–Borel–Carathéodory inequality (Theorem 6) is used to estimate \(\left| \log {\zeta (s)}\right| \) with \(\log {\left| \zeta (s)\right| }\) on the particular circles right of the critical line by using convexity estimates to bound the latter expression, while Hadamard’s three-circles theorem (Theorem 7) then guarantees the “correct” exponent. Given the ensemble of classical ideas, inequality (1) can be generalized without much difficulty to a broader family of functions, e.g. to the Selberg class of functions with a polynomial Euler product, see Sect. 2 for definitions and properties. A similar approach was taken also in [8] where the authors generalized the Lindelöf Hypothesis to some functions in the Selberg class. Our main result is the following theorem.

Theorem 1

Let \(\mathcal {L}(s)\) be an element in the Selberg class of functions with a polynomial Euler product of order m. Then, there exist \(C>0\) and \(c\ge 1\), which depend on \(\mathcal {L}\), and \(T\ge e\) such that

$$\begin{aligned} \log {\left| \mathcal {L}(s)\right| } \le \frac{1}{4}\textrm{d}{}_\mathcal {L}\log {(c|t|)}+m\log {\log {(c|t|)}} + \log ^{+}{C} \end{aligned}$$
(2)

for \(|t|\ge T\), \(\sigma \ge 1/2\) and \(\mathcal {L}(s)\ne 0\), where \(\textrm{d}{}_\mathcal {L}\) is the degree of \(\mathcal {L}(s)\) and \(\log ^{+}{u}\mathrel {\mathop :}=\max \left\{ 0,\log {u}\right\} \) for \(u>0\). For \(C_3>0\) and \(|t|\ge e\), define

$$\begin{aligned} \mathscr {R}\left( C_3,c,t\right) \mathrel {\mathop :}=\left\{ w\in \mathbb {C}:\Re \{w\}>\frac{1}{2}, \left| \Im \{w\}-t\right| \le C_3\log {\log {\left( c(|t|+1)\right) }}+3\right\} , \end{aligned}$$
(3)

where c is from (2). Assume that there exists \(C_3\ge 1\), such that \(\mathcal {L}(z)\ne 0\) for \(z\in \mathscr {R}\left( C_3,c,t\right) \), where |t| is sufficiently large. Then, there exist positive and computable constants \(a_1,b_1\) and \(a_2,b_2\), such that

$$\begin{aligned} \left| \log {\mathcal {L}(s)}\right|\le & {} a_1\left( b_1\log {(c|t|)}\right) ^{2(1-\sigma )}\log {\log {(c|t|)}}, \end{aligned}$$
(4)
$$\begin{aligned} \left| \frac{\mathcal {L}'}{\mathcal {L}}(s)\right|\le & {} a_2\left( b_2\log {(c|t|)}\right) ^{2(1-\sigma )}\left( \log {\log {(c|t|)}}\right) ^2 \end{aligned}$$
(5)

for

$$\begin{aligned} \sigma \in \mathscr {S}\left( A,B,c,t\right) \mathrel {\mathop :}=\left[ \frac{1}{2}+\frac{A}{\log {\log {(c|t|)}}}, 1+\frac{B}{\log {\log {(c|t|)}}}\right] , \end{aligned}$$
(6)

where A and B are some positive constants.

Theorem 1 is contained in Theorem 4 and Corollary 4 where \(a_1,b_1\) and \(a_2,b_2\) are explicitly given as functions in the variables \(\textrm{d}{}_{\mathcal {L}}\), m and C, and “sufficiently large” is precisely defined with known parameters. This enables us to obtain explicit estimates (4) and (5) if we know (2) effectively. Inequality (5) is a consequence of (4) by Cauchy’s integral formula. Note that bounds for \(\left| \log {\mathcal {L}(s)}\right| \) and \(\left| \mathcal {L}'(s)/\mathcal {L}(s)\right| \) can be easily deduced by elementary methods when \(\sigma \ge 1+B/\log {\log {(c|t|)}}\), see Remark 2.

Observe that the condition \(\mathcal {L}(z)\ne 0\) for \(z\in \mathscr {R}\left( C_3,c,t\right) \) is always true under the generalized Riemann hypothesis (GRH), i.e. \(\mathcal {L}(s)\ne 0\) for \(\sigma >1/2\). However, our result is valid under the slightly less restrictive condition which can be viewed as a “local” GRH. Small intervals in the t-aspect come from the radii of the largest circles in the proof being \(\ll \log {\log {|t|}}\).

Estimate (2) is nothing more than a precise form of a well-known convexity result for the Selberg class of functions, see [32, Theorem 6.8]. Our inequality follows by taking the same approach, but with the crucial assumption that \(\mathcal {L}(s)\ll \log ^{\ell }{|t|}\) on the 1-line for some \(\ell >0\) which may depend on \(\mathcal {L}\). Although we are not able to prove this for the full Selberg class, we show that it is true with \(\ell =m\) if we have a polynomial Euler product of order m, see Theorem 3(c). A similar result exists also when axioms on the classical zero-free region and a mild growth condition left to the critical strip are assumed instead of having the axiom on a functional equation, i.e. for the class \(\mathcal {G}\) from [11, Definition 1.2], see Theorem 3(a). We must also emphasize that one could use subconvexity estimates in place of (2), but with the method presented here this would only result in numerical improvements upon \(a_1,b_1\) and \(a_2,b_2\).

Better conditional (RH) estimates than (4) and (5) for the region specified by (6) exist for the Riemann zeta-function, see [33, pp. 383–384], [21, Corollaries 13.14 and 13.16] for a different approach, and [4, Theorem 1] for the latest improvements on constants for the leading terms. To some extent better estimates exist even for general L-functions (in the framework of Iwaniec and Kowalski [16, Chapter 5]) when assuming GRH and the Ramanujan–Petersson conjecture, see [16, Theorems 5.17 and 5.19], and also [6, Theorem 1] when L is entire and satisfies only GRH. The main objective of this paper is thus not to obtain some conditional bounds for fairly large family of L-functions, but rather simultaneously provide their explicit counterparts for three important members: the Riemann, Dirichlet, and Dedekind zeta-functions. It might be interesting to generalize the previously mentioned results to the Selberg class, and then explore possibilities to make them effective.

Corollary 1

(Riemann zeta-function) Let \(\mathcal {L}(s)=\zeta (s)\), where \(s=\sigma +\textrm{i}t\) with \(\sigma >1/2\) and \(|t|\ge 10^4\). Assume that \(\zeta (z)\ne 0\) for \(z\in \mathscr {R}\left( 10^3,1,t\right) \). Then, the following are true:

  1. (a)

    Inequality (4) is valid for \(c=1\), \(a_1=5.44\), \(0.95<b_1<0.951\) and \(\sigma \in \mathscr {S}\left( 0.5,0.5,1,t\right) \).

  2. (b)

    Inequality (5) is valid for \(c=1\), \(a_2=33.281\), \(0.97<b_2<0.971\) and \(\sigma \in \mathscr {S}\left( 1.0051,0.3349,1,t\right) \).

The sets \(\mathscr {R}\) and \(\mathscr {S}\) are defined by (3) and (6), respectively.

Corollary 2

(Dirichlet L-functions) Let \(\mathcal {L}(s)=L\left( s,\chi \right) \), where \(\chi \) is a primitive character modulo \(q\ge 2\) and \(s=\sigma +\textrm{i}t\), where \(\sigma >1/2\) and

$$\begin{aligned} |t|\ge 10450+10^3\log {\log {q}}. \end{aligned}$$

Assume that \(L\left( z,\chi \right) \ne 0\) for \(z\in \mathscr {R}\left( 10^3,q,t\right) \). Then, the following are true:

  1. (a)

    Inequality (4) is valid for \(c=q\), \(a_1=5.44\), \(0.95<b_1<0.951\) and \(\sigma \in \mathscr {S}\left( 0.5,0.5,q,t\right) \).

  2. (b)

    Inequality (5) is valid for \(c=q\), \(a_2=33.281\), \(0.97<b_2<0.971\) and \(\sigma \in \mathscr {S}\left( 1.0051,0.3349,q,t\right) \).

The sets \(\mathscr {R}\) and \(\mathscr {S}\) are defined by (3) and (6), respectively.

Corollary 3

(Dedekind zeta-functions) Let \(\mathcal {L}(s)=\zeta _{\mathbb {K}}(s)\), where \(\mathbb {K}\) is a number field of degree \(n_{\mathbb {K}}\) and discriminant \(\Delta _{\mathbb {K}}\), and \(s=\sigma +\textrm{i}t\), where \(\sigma >1/2\) and

$$\begin{aligned} |t|\ge 9650+10^3\log {\log {\left( 5.545\left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}\right) }}. \end{aligned}$$

Assume that

$$\begin{aligned} \zeta _{\mathbb {K}}(z)\ne 0 \quad \text {for} \quad z\in \mathscr {R}\left( 10^3,5.545\left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}},t\right) . \end{aligned}$$

Then, the following are true:

  1. (a)

    Inequality (4) is valid for \(c=5.545\left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}\), \(a_1=5.44n_{\mathbb {K}}\), \(0.949<b_1<0.95\) and \(\sigma \in \mathscr {S}\left( 0.5,0.5,c,t\right) \).

  2. (b)

    Inequality (5) is valid for \(c=5.545\left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}\), \(a_2=33.281n_{\mathbb {K}}\), \(0.964<b_2<0.965\) and \(\sigma \in \mathscr {S}\left( 1.0051,0.3349,c,t\right) \).

The sets \(\mathscr {R}\) and \(\mathscr {S}\) are defined by (3) and (6), respectively.

Although applications of conditional and effective estimates for L-functions in the critical strip to various number-theoretic problems exist, see [9, p. 20] for instance, results in this direction are quite obscure. Chandee [5] obtained fully explicit bounds for L-functions on the critical line when the analytic conductor is at least of the order \(\exp {\left( \exp (10)\right) }\), while the author [29, Corollary 1] derived a bound for the Riemann zeta-function which is valid for all \(t\ge 2\pi \). Effective upper and lower bounds for \(\zeta (s)\) right to the critical line were provided in [30], thus also covering the region not enclosed by (6), i.e. near the critical line. For explicit estimates on the 1-line see [7, 18, 19].

In [30], the main purpose of having such bounds was establishing conditional (RH) and explicit estimates for the Mertens function \(M(x)=\sum _{n\le x}\mu (n)\) and for the number of k-free numbers, see [30, Theorem 2], where \(x\ge 1\) and \(\mu (n)\) is the Möbius function. Unfortunately, the bounds we have obtained are valid only for very large x, for example

$$\begin{aligned} \left| M(x)\right| \le 0.51x^{0.99}\log {x}, \quad x\ge 10^{10^{4.6}}. \end{aligned}$$

Similarly, estimates for \(m(x)=\sum _{n\le x}\mu (n)/n\) were also provided. A method [30, Remark 1] was proposed to extend validity of estimates of the form \(M(x)\ll x^{\alpha }\) for fixed \(\alpha \in (1/2,1)\) by employing bounds like (4). Here, we are able to prove the following.

Theorem 2

Assume the Riemann hypothesis. For \(x\ge 1\), we then have

$$\begin{aligned} \left| M(x)\right|\le & {} 555.71x^{0.99} + 1.94 \times 10^{14}x^{0.98}, \end{aligned}$$
(7)
$$\begin{aligned} \left| m(x)\right|\le & {} \frac{56126.71}{x^{0.01}} + \frac{9.894 \times 10^{15}}{x^{0.02}}. \end{aligned}$$
(8)

Theorem 2 follows from a more general Theorem 5. Observe that (7) improves on the trivial estimate \(\left| M(x)\right| \le x\) when \(x\ge 10^{714.4}\) and improves the unconditional estimate [26, Theorem 1.1] for \(x\ge 10^{976.8}\), while (8) improves [26, Corollary 1.2] for \(x\ge 10^{1052.1}\). The constants in the second terms in (7) and (8) may be improved by solving a specific computational problem, see Remark 3. It might be interesting to generalize Theorem 2 to the Mertens function in arithmetic progressions using Corollary 2.

The outline of this paper is as follows. In Sect. 2, we revise some properties of functions in the Selberg class and prove inequality (2) by establishing a result on the growth of such functions on the 1-line, see Theorem 3, while also deriving its effective versions for \(\zeta (s)\), \(L(s,\chi )\) and \(\zeta _{\mathbb {K}}(s)\), see Examples 1, 2, and 3. The proofs of Theorem 1 and Corollaries 1, 2, and 3 are provided in Sect. 3, while the proof of Theorem 2 is given in Sect. 4. The Hadamard–Borel–Carathéodory inequality and Hadamard’s three-circles theorem are stated in the appendix.

2 The Selberg Class of Functions

In this section, we are providing a brief overview of some properties of the Selberg class of functions. The emphasis is on studying the growth of such functions on the 1-line (Sect. 2.2) and on deriving an explicit convexity estimates right to the critical line for the Riemann, Dirichlet, and Dedekind zeta-functions (Sect. 2.3).

2.1 Preliminaries

The Selberg class \(\mathcal{S}\mathcal{P}\) of functions with a polynomial Euler product consists of Dirichlet series

$$\begin{aligned} \mathcal {L}(s) = \sum _{n=1}^{\infty } \frac{a(n)}{n^s} \end{aligned}$$
(9)

satisfying the following axioms:

  1. (1)

    Ramanujan hypothesis. We have \(a(n)\ll _{\varepsilon } n^{\varepsilon }\) for any \(\varepsilon >0\).

  2. (2)

    Analytic continuation. There exists \(k\in \mathbb {N}_{0}\) such that \((s-1)^k\mathcal {L}(s)\) is an entire function of finite order.

  3. (3)

    Functional equation. \(\mathcal {L}(s)\) satisfies \(\Lambda _{\mathcal {L}}(s)=\omega \overline{\Lambda _{\mathcal {L}}(1-\bar{s})}\), where

    $$\begin{aligned} \Lambda _{\mathcal {L}}(s)=\mathcal {L}(s)Q^s\prod _{j=1}^{f}\Gamma \left( \lambda _{j}s+\mu _j\right) \end{aligned}$$

    with \(\left( Q,\lambda _j\right) \in \mathbb {R}_{+}^2\), and \(\left( \mu _j,\omega \right) \in \mathbb {C}^2\) with \(\Re \{\mu _j\}\ge 0\) and \(|\omega |=1\).

  4. (4)

    Polynomial Euler product. There exists \(m\in \mathbb {N}\), and for every prime number p there are \(\alpha _j(p)\in \mathbb {C}\), \(1\le j\le m\), such that

    $$\begin{aligned} \mathcal {L}(s) = \prod _{p}\prod _{j=1}^{m}\left( 1-\frac{\alpha _j(p)}{p^s}\right) ^{-1}. \end{aligned}$$

It is well known that axiom (1) implies the absolute convergence of (9) in the half-plane \(\sigma >1\), and that axioms (1) and (4) imply that \(\left| \alpha _j(p)\right| \le 1\) for \(1\le j\le m\) and all prime numbers p, see [32, Lemma 2.2]. Therefore, if \(\mathcal {L}\in \mathcal{S}\mathcal{P}\), then

$$\begin{aligned} \left| \log {\mathcal {L}(s)}\right| = \left| \sum _{p}\sum _{k=1}^{\infty }\sum _{j=1}^{m}\frac{\alpha _j(p)^k}{kp^{ks}}\right| \le m\sum _{p}\sum _{k=1}^{\infty }\frac{1}{kp^{k\sigma _0}} = m\log {\zeta \left( \sigma _0\right) } \end{aligned}$$
(10)

is true for \(\sigma \ge \sigma _0>1\). Inequality (10) implies the following two approximations

$$\begin{aligned} \left| \log {\mathcal {L}(s)}\right|\le & {} m\log {\left( 1+\frac{1}{\sigma _0-1}\right) } \le \frac{m}{\sigma _0-1}, \end{aligned}$$
(11)
$$\begin{aligned} \left| \log {\mathcal {L}(s)}\right|\le & {} m\log {\frac{1}{\sigma _0-1}} + m\gamma \left( \sigma _0-1\right) , \end{aligned}$$
(12)

where \(\gamma \) is the Euler–Mascheroni constant. Estimate (11) follows by comparison with the integral, while (12) is a consequence of [27, Lemma 5.4] and is better than (11) when \(\sigma _0\) is close to 1. Note that \(\mathcal{S}\mathcal{P}\subseteq \mathcal {S}\) where \(\mathcal {S}\) is the classical Selberg class of functions introduced in [28], i.e. axiom (4) is replaced by

$$\begin{aligned} \mathcal {L}(s) = \prod _{p}\exp {\left( \sum _{k=1}^{\infty }\frac{b\left( p^k\right) }{p^{ks}}\right) } \end{aligned}$$

where the coefficients \(b\left( p^k\right) \) satisfy \(b\left( p^k\right) \ll p^{k\theta }\) for some \(0\le \theta <1/2\). It is conjectured that \(\mathcal{S}\mathcal{P}=\mathcal {S}\).

The degree of \(\mathcal {L}\in \mathcal {S}\) is defined by \(\textrm{d}{}_{\mathcal {L}}=2\sum _{j=1}^{f}\lambda _j\). Since \(N_{\mathcal {L}}(T)\sim \frac{1}{\pi }\textrm{d}{}_{\mathcal {L}}T\log {T}\), where \(N_{\mathcal {L}}(T)\) counts the number of zerosFootnote 1 of \(\mathcal {L}(s)\) with \(\sigma \in [0,1]\) and \(|t|\le T\), it follows that \(\textrm{d}{}_{\mathcal {L}}\) is well defined although parameters from axiom (3) are not unique. Note that \(\textrm{d}{}_{1}=0\) and \(\textrm{d}{}_{\zeta }=1\). It is known [32, Theorem 6.1] that \(\textrm{d}{}_{\mathcal {L}}\ge 1\) for every \(\mathcal {L}\in \mathcal {S}{\setminus }\{1\}\), and it is conjectured that \(\textrm{d}{}_{\mathcal {L}}\) is always a positive integer. Kaczorowski and Perelli proved that \(\zeta (s)\) and shifts \(L(s+\textrm{i}\theta ,\chi )\), \(\theta \in \mathbb {R}\), of Dirichlet L-functions (\(\sigma >1\))

$$\begin{aligned} L\left( s,\chi \right) = \sum _{n=1}^{\infty }\frac{\chi (n)}{n^s} = \prod _{p}\left( 1-\frac{\chi (p)}{p^s}\right) ^{-1} \end{aligned}$$

attached to a primitive character \(\chi \) modulo \(q\ge 2\), are the only functions in \(\mathcal {S}\) with degree 1, see [31] for a simplified proof. Important examples are also Dedekind zeta-functions (\(\sigma >1\))

$$\begin{aligned} \zeta _{\mathbb {K}}(s) = \sum _{\mathfrak {a}}\frac{1}{N(\mathfrak {a})^s} = \prod _{\mathfrak {p}}\left( 1-\frac{1}{N(\mathfrak {p})^s}\right) ^{-1} = \prod _{p}\prod _{\begin{array}{c} j=1 \\ \mathfrak {p}_j|(p) \end{array}}^{r}\left( 1-\frac{1}{p^{sf_j}}\right) ^{-1}, \end{aligned}$$

where \(\mathbb {K}\) is a number field, \(N(\cdot )\) is the norm of an ideal, \(\mathfrak {a}\) runs through all non-zero ideals and \(\mathfrak {p}\) runs through all prime ideals of the ring of integers of \(\mathbb {K}\). The last equality follows because any rational prime number p has a unique factorization \((p) = \prod _{j=1}^{r}\mathfrak {p}_{j}^{e_j}\) with \(N\left( \mathfrak {p}_{j}\right) =p^{f_j}\) and \(\sum _{j=1}^{r}e_jf_j=n_{\mathbb {K}}\mathrel {\mathop :}=\left[ \mathbb {K}:\mathbb {Q}\right] \), where the positive integers \(e_j\), \(f_j\) and r depend on p. Therefore, \(r\le n_{\mathbb {K}}\), which implies a polynomial Euler product representation for \(m=n_{\mathbb {K}}\). We have that \(\zeta _{\mathbb {K}}\) belongs to \(\mathcal{S}\mathcal{P}\) and \(\textrm{d}{}_{\zeta _{\mathbb {K}}}=n_{\mathbb {K}}\). Observe also that \(\zeta _{\mathbb {Q}}(s)=\zeta (s)\).

The functional equation from axiom (3) can be written as \(\mathcal {L}(s)=\Delta _{\mathcal {L}}(s)\overline{\mathcal {L}\left( 1-\bar{s}\right) }\), where

$$\begin{aligned} \Delta _{\mathcal {L}}(s) \mathrel {\mathop :}=\omega Q^{1-2s}\prod _{j=1}^{f}\frac{\Gamma \left( \lambda _j(1-s)+\overline{\mu _j}\right) }{\Gamma \left( \lambda _{j}s+\mu _j\right) }. \end{aligned}$$

Taking \(\mathcal {L}\in \mathcal {S}\), we can use Stirling’s formula to prove

$$\begin{aligned} \Delta _{\mathcal {L}}(s)\ll |t|^{\textrm{d}{}_{\mathcal {L}}\left( \frac{1}{2}-\sigma \right) }, \end{aligned}$$
(13)

where this estimate is uniform in \(\sigma \in \left[ \sigma _1,\sigma _2\right] \) for fixed \(\sigma _1\le \sigma _2\), see [32, Lemma 6.7]. It is possible to make (13) uniform also in \(\mathcal {L}\) by means of the data of the functional equation, but such an approach is not needed in the present paper.

2.2 On the Growth of \(\mathcal {L}(s)\) on the 1-Line

It is convenient to introduce an additional axiom which concerns the growth of \(\mathcal {L}\left( 1+\textrm{i}t\right) \) when \(\mathcal {L}\in \mathcal {S}\) and \(|t|\rightarrow \infty \).

  1. (5)

    Growth on the 1-line. \(\mathcal {L}\left( 1+\textrm{i}t\right) \ll \log ^{\ell }|t|\) for some \(\ell >0\).

In the case of the Riemann zeta-function, it is a standard result that we can take \(\ell =1\), while a substantial improvement to \(\ell =2/3\) requires techniques from the proof of the Vinogradov–Korobov’s zero-free region, see [15, Chapter 6]. Note that the former result can be proved using the approximate functional equation for \(\zeta (s)\). A similar approach is also used in the proof of Theorem 3(b).

Dixit and Mahatab introduced in [11, Definition 1.2] a new class of functions \(\mathcal {G}\). We say that \(\mathcal {L}\in \mathcal {G}\), if the series (9) is absolutely convergent for \(\sigma >1\), \(a(1)=1\), and \(\mathcal {L}\) satisfies beside axioms (2) and (4) also the following two axioms:

  1. (6)

    Zero-free region. There exists \(c_{\mathcal {L}}>0\) such that \(\mathcal {L}\) has no zeros in the region

    $$\begin{aligned} \left\{ z\in \mathbb {C}:\Re \{z\}\ge 1-\frac{c_{\mathcal {L}}}{\log {\left( \left| \Im \{z\}\right| +2\right) }}\right\} , \end{aligned}$$

    except the possible Siegel zero, i.e. real exceptional zero of \(\mathcal {L}\) in the neighbourhood of 1.

  2. (7)

    Growth condition. Define \(\mu _{\mathcal {L}}^{*}(\sigma )\mathrel {\mathop :}=\inf \left\{ \lambda >0:\mathcal {L}(\sigma +\textrm{i}t)\ll _{\sigma }|t|^{\lambda }\right\} \). Then, \(\mu _{\mathcal {L}}^{*}(\sigma ) \ll 1-2\sigma \) uniformly for \(\sigma <0\).

Observe that here class \(\mathcal {G}\) does not require a functional equation; axiom (3) implies axiom (7), but the latter is sufficient to show that then \(\mathcal {L}(s)\) is polynomially bounded in vertical strips by using the Phragmén–Lindelöf principle. It is expected that \(\mathcal {S}\subseteq \mathcal {G}\).

The next theorem explores possible connections between classes \(\mathcal {G}\), \(\mathcal {S}\), \(\mathcal{S}\mathcal{P}\) and axiom (5). As usual, \(d_{\alpha }(n)\) denotes the number of ways positive integer n can be written as a product of \(\alpha \ge 2\) factors, and we extend this to \(d_{1}(n)\equiv 1\).

Theorem 3

The following are true:

  1. (a)

    Let \(\mathcal {L}\in \mathcal {G}\) and take \(\varepsilon >0\). Then, \(\mathcal {L}\) satisfies axiom (5) with \(\ell =m+\varepsilon \).

  2. (b)

    Let \(\mathcal {L}\in \mathcal {S}\) and assume \(a(n)\ll d_{\alpha }(n)\) for some positive integer \(\alpha \). Then, \(\mathcal {L}\) satisfies axiom (5) with \(\ell =\alpha \).

  3. (c)

    Let \(\mathcal {L}\in \mathcal{S}\mathcal{P}\). Then, \(\mathcal {L}\) satisfies axiom (5) with \(\ell =m\).

Proof

First, we are going to prove the assertion (a) by following the method from [11]. Take \(\mathcal {L}\in \mathcal {G}\). Let \(X\ge 2\), \(\sigma >1\) and

$$\begin{aligned} \mathcal {L}\left( s;X\right) \mathrel {\mathop :}=\prod _{p\le X}\prod _{j=1}^{m}\left( 1-\frac{\alpha _{j}(p)}{p^s}\right) ^{-1}, \quad \log {\mathcal {L}(s)} = \sum _{n=1}^{\infty }\frac{b_{\mathcal {L}}(n)}{n^s}. \end{aligned}$$

Axiom (4) asserts that \(b_{\mathcal {L}}(n)=0\) if \(n\ne p^k\) and \(\left| b_{\mathcal {L}}(n)\right| \le m\) otherwise. For \(\sigma \ge 1\), it follows that

$$\begin{aligned} \log {\mathcal {L}\left( s;X\right) }= & {} \sum _{p\le X}\sum _{k=1}^{\infty }\frac{b_{\mathcal {L}}\left( p^k\right) }{p^{ks}} \nonumber \\= & {} \sum _{n\le X}\frac{b_{\mathcal {L}}(n)}{n^s} + \left( \sum _{\sqrt{X}<p\le X}\sum _{p^k>X} + \sum _{p\le \sqrt{X}}\sum _{p^k>X}\right) \frac{b_{\mathcal {L}}\left( p^k\right) }{p^{ks}} \nonumber \\= & {} \sum _{n\le X}\frac{b_{\mathcal {L}}(n)}{n^s} + O\left( \sum _{\sqrt{X}<p\le X}\sum _{k=2}^{\infty }\frac{1}{p^k} + \sum _{p\le \sqrt{X}}\frac{1}{X}\right) \nonumber \\= & {} \sum _{n\le X}\frac{b_{\mathcal {L}}(n)}{n^s} + O\left( \frac{1}{\sqrt{X}}\right) \end{aligned}$$
(14)

since \(\left| p^{ks}\right| =p^{k\sigma }\ge p^k\).

For \(t\ge 2\), \(\alpha >0\) and \(\varepsilon >0\), define

$$\begin{aligned} \sigma _1 \mathrel {\mathop :}=\frac{1}{\alpha \log {t}}, \quad \sigma _2 \mathrel {\mathop :}=\frac{1}{\left( \log {t}\right) ^{1+\varepsilon /m}}. \end{aligned}$$

By Perron’s formula, we have

$$\begin{aligned} \sum _{n\le X}\frac{b_{\mathcal {L}}(n)}{n^{1+\textrm{i}t}} = \frac{1}{2\pi \textrm{i}}\int _{\sigma _2-\frac{\textrm{i}t}{2}}^{\sigma _2+\frac{\textrm{i}t}{2}} \log {\mathcal {L}\left( 1+\textrm{i}t+z\right) }\frac{X^{z}}{z}\textrm{d}{z} + O\left( \frac{X^{\sigma _2}}{t\sigma _2}+\frac{\log {X}}{t}+\frac{1}{X}\right) . \end{aligned}$$
(15)

Let

$$\begin{aligned} \mathscr {C} \mathrel {\mathop :}=\left\{ z\in \mathbb {C}:1-\sigma _1\le \Re \{z\}\le 1+\sigma _2, \frac{t}{2}\le \Im \{z\}\le \frac{3t}{2}\right\} . \end{aligned}$$

By axiom (6) there exist \(\alpha \) and \(t_0>0\) such that there are no zeros of \(\mathcal {L}(s)\) in a neighbourhood of \(\mathscr {C}\) for \(t\ge t_0\). Moreover, one can use Theorem 6 together with axiom (7) and inequality (10) to prove that \(\log {\mathcal {L}(z)}\ll \log {t}\) for \(z\in \partial \mathscr {C}\). Take \(X=\exp {\left( \left( \log {t}\right) ^{1+\varepsilon /m}\right) }\). By (14), (15) and Cauchy’s formula, we then have

$$\begin{aligned}&\log {\mathcal {L}\left( 1+\textrm{i}t\right) } = \log {\mathcal {L}\left( 1+\textrm{i}t;X\right) } + O(1) \\&\quad + \frac{1}{2\pi \textrm{i}}\left( \int _{\sigma _2+\frac{\textrm{i}t}{2}}^{-\sigma _1+\frac{\textrm{i}t}{2}}+\int _{-\sigma _1+\frac{\textrm{i}t}{2}}^{-\sigma _1-\frac{\textrm{i}t}{2}}+\int _{-\sigma _1-\frac{\textrm{i}t}{2}}^{\sigma _2-\frac{\textrm{i}t}{2}}\right) \log {\mathcal {L}\left( 1+\textrm{i}t+z\right) }\frac{X^{z}}{z}\textrm{d}{z}. \end{aligned}$$

We obtain

$$\begin{aligned} \int _{\sigma _2+\frac{\textrm{i}t}{2}}^{-\sigma _1+\frac{\textrm{i}t}{2}} \log {\mathcal {L}\left( 1+\textrm{i}t+z\right) }\frac{X^{z}}{z}\textrm{d}{z} \ll \frac{X^{\sigma _2}\log {t}}{t\log {X}} \ll 1 \end{aligned}$$

with the same result also for the third integral while the second integral may be bounded as

$$\begin{aligned} \int _{-\sigma _1+\frac{\textrm{i}t}{2}}^{-\sigma _1-\frac{\textrm{i}t}{2}} \log {\mathcal {L}\left( 1+\textrm{i}t+z\right) }\frac{X^{z}}{z}\textrm{d}{z} \ll X^{-\sigma _1}\log ^{2}{t} = \frac{\log ^{2}{t}}{\exp {\left( \alpha ^{-1}\left( \log {t}\right) ^{\varepsilon /m}\right) }} \ll 1. \end{aligned}$$

Therefore, \(\log {\mathcal {L}\left( 1+\textrm{i}t\right) }=\log {\mathcal {L}\left( 1+\textrm{i}t;X\right) } + O(1)\). Since

$$\begin{aligned} \left| \mathcal {L}\left( 1+\textrm{i}t;X\right) \right| \le \prod _{p\le X}\left( 1-\frac{1}{p}\right) ^{-m} \ll \log ^{m}{X} \end{aligned}$$

by Mertens’ third theorem, it follows that

$$\begin{aligned} \mathcal {L}\left( 1+\textrm{i}t\right) \ll \left| \mathcal {L}\left( 1+\textrm{i}t;X\right) \right| \ll \log ^{m}{X} = \left( \log {t}\right) ^{m+\varepsilon }. \end{aligned}$$

In a similar way, we can obtain such an estimate when t is negative. The proof of Theorem 3(a) is thus complete.

We are going to prove the assertion (b). Take \(\mathcal {L}\in \mathcal {S}\), \(x\ge 1\) and \(t_0>0\) sufficiently large. We can assume that \(\mathcal {L}\not \equiv 1\) since otherwise the result is trivial. For \(|t|\ge t_0\), we have

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{a(n)}{n^{1+\textrm{i}t}}e^{-\left( \frac{n}{x}\right) ^{\log {|t|}}}&= \frac{1}{2\pi \textrm{i}}\int _{2-\textrm{i}\infty }^{2+\textrm{i}\infty }\frac{x^z}{z}\mathcal {L}\left( 1+\textrm{i}t+z\right) \Gamma \left( 1+\frac{z}{\log {|t|}}\right) \textrm{d}{z} \\&= \frac{1}{2\pi \textrm{i}}\int _{-\frac{3}{2}-\textrm{i}\infty }^{-\frac{3}{2}+\textrm{i}\infty }\frac{x^z \overline{\mathcal {L}\left( \textrm{i}t-\bar{z}\right) }\Delta _{\mathcal {L}}\left( 1+\textrm{i}t+z\right) \Gamma \left( 1+\frac{z}{\log {|t|}}\right) }{z}\textrm{d}{z} \\&\quad + \mathcal {L}\left( 1+\textrm{i}t\right) + \frac{\textrm{i}x^{-\textrm{i}t}}{t}\Gamma \left( 1-\frac{\textrm{i}t}{\log {|t|}}\right) \textrm{Res}\left( \mathcal {L}(s),1\right) . \end{aligned}$$

The first equality follows from the classical Mellin integral, while the second follows by moving the line of integration to \(\Re \{z\}=-3/2\), using the functional equation, and detecting two poles at \(z=0\) and \(z=-\textrm{i}t\) of the integrand which are inside the contour. It is clear that the second residue is O(1). We are going to demonstrate that this is also true for the second integral in the last expression if we take x large enough. Denote this integral by \(\mathcal {I}\) and let \(z=-3/2+\textrm{i}u\), \(u\in \mathbb {R}\). Then,

$$\begin{aligned} \Delta _{\mathcal {L}}\left( 1+\textrm{i}t+z\right) \ll \left( |u+t|+1\right) ^{\textrm{d}{}_{\mathcal {L}}} \ll \left\{ \begin{array}{ll} |t|^{\textrm{d}{}_{\mathcal {L}}}, &{} |u|\le \log {|t|}, \\ |t|^{\textrm{d}{}_{\mathcal {L}}}|u|^{\textrm{d}{}_{\mathcal {L}}}, &{} |u|>\log {|t|}, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \Gamma \left( 1+\frac{z}{\log {|t|}}\right) \ll \left\{ \begin{array}{ll} 1, &{} |u|\le \log {|t|}, \\ \left( \frac{|u|}{\log {|t|}}\right) ^{\frac{1}{2}}\exp {\left( -\frac{|u|}{\log {|t|}}\right) }, &{} |u|>\log {|t|}, \end{array} \right. \end{aligned}$$

while the implied constants are uniform in u and t. Obviously, \(\mathcal {L}\left( \textrm{i}t-\bar{z}\right) \ll 1\). Splitting the range of integration in \(\mathcal {I}\) into two parts, \(|u|\le \log {|t|}\) and \(|u|>\log {|t|}\), we obtain

$$\begin{aligned} \mathcal {I} \ll x^{-\frac{3}{2}}|t|^{\textrm{d}{}_{\mathcal {L}}}\left( \log {|t|}+\Gamma \left( \frac{3}{2}+\textrm{d}{}_{\mathcal {L}},1\right) \log ^{\textrm{d}{}_{\mathcal {L}}}{|t|}\right) \ll x^{-\frac{3}{2}}|t|^{\textrm{d}{}_{\mathcal {L}}}\log ^{\textrm{d}{}_{\mathcal {L}}}{|t|}. \end{aligned}$$

From the last expression, we can see that \(\mathcal {I}=O(1)\) if \(x=|t|^{\textrm{d}{}_{\mathcal {L}}}\). With such choice for x, we also have

$$\begin{aligned} \sum _{n>ex}\frac{a(n)}{n^{1+\textrm{i}t}}e^{-\left( \frac{n}{x}\right) ^{\log {|t|}}}&\ll \sum _{n>ex}e^{-\left( \frac{n}{x}\right) ^{\log {|t|}}} \le e^{-|t|} + x\int _{e}^{\infty }e^{-u^{\log {|t|}}}\textrm{d}{u} \\&\ll e^{-|t|} + x|t|^{-\log {|t|}} \ll 1, \end{aligned}$$

where we used \(a(n)\ll n\) and \(\log ^{2}{|t|}+u\le u^{\log {|t|}}\), the last inequality valid for \(u\ge 2\) and \(\log {|t|}\ge 5\). All of these finally imply

$$\begin{aligned} \mathcal {L}\left( 1+\textrm{i}t\right) = \sum _{n\le e|t|^{\textrm{d}{}_{\mathcal {L}}}}\frac{a(n)}{n^{1+\textrm{i}t}}e^{-\left( n/|t|^{\textrm{d}{}_{\mathcal {L}}}\right) ^{\log {|t|}}} + O(1). \end{aligned}$$

Since \(a(n)\ll d_{\alpha }(n)\) by the assumption, and \(\sum _{n\le X}d_{\alpha }(n)\ll X\left( \log {X}\right) ^{\alpha -1}\), it follows that

$$\begin{aligned} \mathcal {L}\left( 1+\textrm{i}t\right) \ll \sum _{n\le e|t|^{\textrm{d}{}_{\mathcal {L}}}}\frac{d_{\alpha }(n)}{n} \ll \log ^{\alpha }{|t|} \end{aligned}$$

by partial summation. The proof of statement (b) is thus complete.

The proof of statement (c) now easily follows from the assertion (b) since one can observe that the estimate for the local roots \(\left| \alpha _{j}(p)\right| \le 1\) implies \(\left| a(n)\right| \le d_{m}(n)\), see [32, Lemma 2.2], and the former is true for functions in \(\mathcal{S}\mathcal{P}\). \(\square \)

Our proof of Theorem 3(b) follows a similar approach as the proof of a smooth version of the approximate functional equation for \(\zeta (s)\), see [15, Theorem 4.4], and also [20, Proposition 2.3] for a generalization to the Selberg class and [16, Theorem 5.3] for a generalization to L-functions. In correspondence with these functions, our condition \(a(n)\ll d_{\alpha }(n)\) can be viewed as an analogue to the Ramanujan–Petersson conjecture. However, for our purpose, we do not require a complete result, so the proof can be simplified.

2.3 Convexity Estimates for \(\mathcal {L}(s)\)

Assuming axiom (5), it is easy to prove the precise form of the convexity-type result for \(\mathcal {L}(s)\).

Proposition 1

Take \(\sigma _0<0\), \(\mathcal {L}\in \mathcal {S}\) and assume that \(\mathcal {L}\) also satisfies axiom (5). Then,

$$\begin{aligned} \mathcal {L}(s) \ll \left\{ \begin{array}{ll} |t|^{\textrm{d}{}_{\mathcal {L}}\left( \frac{1}{2}-\sigma \right) }\log ^{\ell }|t|, &{} \sigma _0\le \sigma < 0, \\ |t|^{\frac{1}{2}\textrm{d}{}_{\mathcal {L}}(1-\sigma )}\log ^{\ell }|t|, &{} 0\le \sigma \le 1, \\ \log ^{\ell }|t|, &{} \sigma >1, \end{array} \right. \end{aligned}$$

where the implied constants are uniform in \(\sigma \).

Proof

Axiom (5), estimate (13) and the functional equation imply

$$\begin{aligned} \mathcal {L}(\textrm{i}t)\ll |t|^{\frac{1}{2}\textrm{d}{}_{\mathcal {L}}}\log ^{\ell }{|t|}. \end{aligned}$$

Also, the estimate for \(\sigma \in \left[ \sigma _0,0\right) \) follows from the estimate for \(\sigma >1\) and the functional equation, so it remains to prove the bounds for \(\sigma \ge 0\).

For \(\sigma >-1\), define

$$\begin{aligned} f_{\mathcal {L}}(s) \mathrel {\mathop :}=\frac{(s-1)^{k+\textrm{d}{}_{\mathcal {L}}}\mathcal {L}(s)}{(s+1)^{\textrm{d}{}_{\mathcal {L}}(3-\sigma )/2+k}\log ^{\ell }(s+2)}, \quad g_{\mathcal {L}}(s) \mathrel {\mathop :}=\frac{(s-1)^{k}\mathcal {L}(s)}{(s+1)^{k}\log ^{\ell }(s+2)}, \end{aligned}$$

where k is from axiom (2). Then, \(f_{\mathcal {L}}(s)\) and \(g_{\mathcal {L}}(s)\) are holomorphic functions of finite order in the half-plane \(\left\{ z\in \mathbb {C}:\Re \{z\}>-1\right\} \).

Since \(\left| f_{\mathcal {L}}(1+\textrm{i}t)\right| \) and \(\left| f_{\mathcal {L}}(\textrm{i}t)\right| \) are bounded for all \(t\in \mathbb {R}\), the Phragmén–Lindelöf theorem implies that also \(\left| f_{\mathcal {L}}(s)\right| \) is bounded for \(\sigma \in [0,1]\) and \(t\in \mathbb {R}\). This proves the first estimate.

Trivially, \(\mathcal {L}(s)\ll \log ^{\ell }|t|\) for \(\sigma \ge 2\). As before, because \(\left| g_{\mathcal {L}}(1+\textrm{i}t)\right| \) and \(\left| g_{\mathcal {L}}(2+\textrm{i}t)\right| \) are bounded for all \(t\in \mathbb {R}\), this implies that also \(\left| g_{\mathcal {L}}(s)\right| \) is bounded for all \(\sigma \in \left[ 1,2\right] \) and \(t\in \mathbb {R}\). The proof is thus complete. \(\square \)

Remark 1

Note that inequality (2) from Theorem 1 immediately follows from Proposition 1 and Theorem 3(c).

We are going to provide numerical values for the constants C, c and T from Theorem 1 in the case when \(\mathcal {L}(s)\) is \(\zeta (s)\), \(L(s,\chi )\), and \(\zeta _{\mathbb {K}}(s)\).

Example 1

(Riemann zeta-function) For \(\mathcal {L}(s)=\zeta (s)\), we have \(\textrm{d}{}_\mathcal {L}=m=1\). Let \(|t|\ge 50\). Backlund [1, Equations (54) and (56)] proved that \(\left| \zeta (s)\right| \le \log {|t|}\) for \(\sigma >1\), and

$$\begin{aligned} \left| \zeta (s)\right| \le \frac{t^2}{t^2-4}\left( \frac{|t|}{2\pi }\right) ^{\frac{1-\sigma }{2}}\log {|t|} \end{aligned}$$

for \(\sigma \in [0,1]\). It follows that inequality (2) is valid for the values \(C=c=1\) and \(T=50\).

Example 2

(Dirichlet L-functions) Let \(\chi \) be a primitive character modulo \(q\ge 2\). For \(\mathcal {L}(s)=L(s,\chi )\) we have \(\textrm{d}{}_\mathcal {L}=m=1\). Rademacher [25, Theorem 3] proved that

$$\begin{aligned} \left| L(s,\chi )\right| \le \left( \frac{q|1+s|}{2\pi }\right) ^{\frac{1+\eta -\sigma }{2}}\zeta (1+\eta ) \end{aligned}$$

for \(\sigma \in [-\eta ,1+\eta ]\) and \(\eta \in (0,1/2]\). Take \(\eta =\alpha /\log {(q|t|)}\), \(\alpha \ge 1\), \(\sigma \in [1/2,1+\eta ]\) and \(|t|\ge t_0\ge e^{2\alpha }\). Since

$$\begin{aligned}{} & {} \zeta (1+\eta ) \le \frac{1}{\eta }e^{\gamma \eta } \le \frac{1}{\alpha }\exp {\left( \frac{\gamma \alpha }{\log {t_0}}\right) }\log {(q|t|)},\nonumber \\ 1{} & {} \le \frac{q|1+s|}{2\pi } \le \frac{1}{2\pi }\sqrt{1+\left( \frac{2+\frac{\alpha }{\log {t_0}}}{t_0}\right) ^{2}}q|t|,\\ 0{} & {} \le \frac{1+\eta -\sigma }{2}\le \frac{1}{4}+\frac{\alpha }{2\log {(q|t|)}}\nonumber \end{aligned}$$
(16)

with the first set of inequalities true by (12), it follows that

$$\begin{aligned} \left| L(s,\chi )\right| {}{} & {} \le \frac{1}{\alpha }\exp {\left( \alpha \left( \frac{1}{2}+\frac{\gamma }{\log {t_0}}\right) \right) }\\ {}{} & {} \times \left( \frac{1}{2\pi }\sqrt{1+\left( \frac{2+\frac{\alpha }{\log {t_0}}}{t_0}\right) ^{2}}\right) ^{\frac{1}{4}}\left( q|t|\right) ^{\frac{1}{4}}\log {(q|t|)}. \end{aligned}$$

Take \(t_0=7778\) and \(\alpha =1.8\). Then, the last inequality implies that \(\left| L(s,\chi )\right| \le \left( q|t|\right) ^{\frac{1}{4}}\log {(q|t|)}\) for \(1/2\le \sigma \le 1+1.8/\log {(q|t|)}\) and \(|t|\ge 7778\). The same bound holds by (16) also for \(\sigma \ge 1+1.8/\log {(q|t|)}\). Therefore, inequality (2) is valid for the values \(C=1\), \(c=q\) and \(T=7778\).

Example 3

(Dedekind zeta-functions) Let \(\mathbb {K}\) be a number field of degree \(n_{\mathbb {K}}\) and discriminant \(\Delta _{\mathbb {K}}\). For \(\mathcal {L}(s)=\zeta _{\mathbb {K}}(s)\) we have \(\textrm{d}{}_\mathcal {L}=m=n_{\mathbb {K}}\). Note that \(\left| \Delta _{\mathbb {K}}\right| \ge 1\). The displayed equation above Theorem 4 from [25] implies that

$$\begin{aligned} \left| \zeta _{\mathbb {K}}(s)\right| \le \frac{1+\eta }{1-\eta }\left| \frac{1+s}{1-s}\right| \left( \left| \Delta _{\mathbb {K}}\right| \left( \frac{|1+s|}{2\pi }\right) ^{n_{\mathbb {K}}}\right) ^{\frac{1+\eta -\sigma }{2}}\zeta (1+\eta )^{n_{\mathbb {K}}} \end{aligned}$$

for \(\sigma \in [-\eta ,1+\eta ]\), \(\eta \in (0,1/2]\) and \(s\ne 1\). Take \(\eta =\alpha /\log {\left( \left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}|t|\right) }\), \(\alpha \ge 1\), \(\sigma \in [1/2,1+\eta ]\) and \(|t|\ge t_0\ge e^{2\alpha }\). A similar procedure as in Example 2 guarantees

$$\begin{aligned} \left| \zeta _{\mathbb {K}}(s)\right|&\le \frac{1}{\left( 2\pi \right) ^{\frac{1}{4}}}\frac{1+\frac{\alpha }{\log {t_0}}}{1-\frac{\alpha }{\log {t_0}}}\left( 1+\left( \frac{2+\frac{\alpha }{\log {t_0}}}{t_0}\right) ^2\right) ^{\frac{5}{8}} \\&\quad \times \left( \frac{1}{\alpha }\exp {\left( \alpha \left( \frac{1}{2}+\frac{\gamma }{\log {t_0}}\right) \right) }\right) ^{n_{\mathbb {K}}}\left( \left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}|t|\right) ^{\frac{1}{4}n_{\mathbb {K}}}\\&\quad \times \log ^{n_{\mathbb {K}}}{\left( \left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}|t|\right) }. \end{aligned}$$

Take \(t_0=7778\) and \(\alpha =1.8\), and let \(c=5.545\left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}\). Then, the last inequality implies that \(\left| \zeta _{\mathbb {K}}(s)\right| \le \left( c|t|\right) ^{\frac{1}{4}n_{\mathbb {K}}}\log ^{n_{\mathbb {K}}}{\left( c|t|\right) }\) for \(1/2\le \sigma \le 1+1.8/\log {\left( c|t|\right) }\) and \(|t|\ge 7778\). The same bound holds also for \(\sigma \ge 1+1.8/\log {\left( c|t|\right) }\). We deduce that inequality (2) is valid for the values \(C=1\), \(c=5.545\left| \Delta _{\mathbb {K}}\right| ^{1/n_{\mathbb {K}}}\) and \(T=7778\).

3 Proof of Theorem 1 and Its Corollaries

In this section, we prove the estimates on \(\log {\mathcal {L}(s)}\) and \(\mathcal {L}'(s)/\mathcal {L}(s)\) from Theorem 1 by explicitly expressing the corresponding constants as functions in variables from our convexity estimate (2), see Theorem 4 and Corollary 4. Next, we use these results in combination with Examples 1, 2 and 3 to prove Corollaries 1, 2 and 3.

First, we will isolate a result which compares \(\left| \log {\mathcal {L}(z)}\right| \) with the estimate (2) on some particular circles by means of Theorem 7.

Lemma 1

Take \(\mathcal {L}\in \mathcal{S}\mathcal{P}\). Let \(|t'|\ge t_0'\ge \max \left\{ T+1,\exp {\left( e^2\right) }\right\} \) where T is from Theorem 1, \(0<C_1\le 1\) and \(0<\delta \le 1/2\). Assume that \(\mathcal {L}(z)\ne 0\) for \(\Re \{z\}>1/2\) and \(\left| \Im \{z\}-t'\right| \le 1\). Define

$$\begin{aligned} \mathscr {D}\left( C_1,\delta ,t'\right) \mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\left| 1+\frac{C_1}{\log {\log {|t'|}}}+\textrm{i}t' - z\right| \le \frac{1}{2}+\frac{C_1}{\log {\log {|t'|}}}-\delta \right\} . \end{aligned}$$
(17)

Then,

$$\begin{aligned} \left| \log {\mathcal {L}(z)}\right| \le \frac{1}{\delta }K\log {\left( c\left( |t'|+1\right) \right) } \end{aligned}$$
(18)

for \(z\in \mathscr {D}\left( C_1,\delta ,t'\right) \), where \(c\ge 1\) is from (2),

$$\begin{aligned}{} & {} K\left( \textrm{d}{}_{\mathcal {L}},m,C,C_1,t_0'\right) \mathrel {\mathop :}=\frac{1}{4}\textrm{d}{}_{\mathcal {L}} + \frac{C_1\textrm{d}{}_{\mathcal {L}}}{2\log {\log {t_0'}}} + \left( 1+\frac{2C_1}{\log {\log {t_0'}}}\right) \nonumber \\{} & {} \quad \times \left( \frac{m\log {\log {t_0'}}}{\log {t_0'}}+\frac{m}{\log {t_0'}}\left( \log {\log {\log {t_0'}}}+\log {\frac{1}{C_1}}+\frac{\gamma C_1}{\log {\log {t_0'}}}\right) \right. \nonumber \\{} & {} \qquad \left. +\frac{\log ^{+}{C}}{\log {t_0'}}\right) , \end{aligned}$$
(19)

m is from axiom (4) and C is from inequality (2).

Proof

Let \(\lambda \in (0,1)\) and define

$$\begin{aligned} \mathscr {D}_0 \mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\left| 1+\frac{C_1}{\log {\log {|t'|}}}+\textrm{i}t' - z\right| \le \frac{1}{2}+\frac{C_1}{\log {\log {|t'|}}}-\lambda \delta \right\} . \end{aligned}$$

Observe that \(\mathscr {D}=\mathscr {D}\left( C_1,\delta ,t'\right) \) and \(\mathscr {D}_0\) are closed discs with the same centre, and

$$\begin{aligned} \mathscr {D} \subseteq \mathscr {D}_0 \subset \left\{ z\in \mathbb {C}:\Re \{z\}>\frac{1}{2}, \left| \Im \{z\}-t'\right| <1\right\} . \end{aligned}$$

Since \(\log {\mathcal {L}(z)}\) is a holomorphic function on the last domain, Theorem 6 implies

$$\begin{aligned}{} & {} \max _{z\in \partial {\mathscr {D}}}\left\{ \left| \log {\mathcal {L}(z)}\right| \right\} \le \frac{1}{(1-\lambda )\delta }\left( 1+\frac{2C_1}{\log {\log {|t'|}}}-2\delta \right) \max _{z\in \partial {\mathscr {D}_0}}\left\{ \Re \left\{ \log {\mathcal {L}(z)}\right\} \right\} \nonumber \\ {}{} & {} \quad + \frac{1}{(1-\lambda )\delta }\left( 1+\frac{2C_1}{\log {\log {|t'|}}}-(1+\lambda )\delta \right) \nonumber \\ {}{} & {} \quad \times \left| \log {\mathcal {L}\left( 1+\frac{C_1}{\log {\log {|t'|}}}+\text {i}t'\right) }\right| . \end{aligned}$$
(20)

By (2), we have

$$\begin{aligned} \max _{z\in \partial {\mathscr {D}_0}}\left\{ \Re \left\{ \log {\mathcal {L}(z)}\right\} \right\} \le \left( \frac{1}{4}\textrm{d}{}_\mathcal {L}+\frac{m\log {\log {t_0'}}}{\log {t_0'}}\right) \log {\left( c\left( |t'|+1\right) \right) } + \log ^{+}{C}, \end{aligned}$$
(21)

while (12) guarantees that

$$\begin{aligned} \left| \log {\mathcal {L}\left( 1+\frac{C_1}{\log {\log {|t'|}}}+\textrm{i}t'\right) }\right| \le m\left( \log {\log {\log {|t'|}}}+\log {\frac{1}{C_1}}+\frac{\gamma C_1}{\log {\log {|t'|}}}\right) . \end{aligned}$$
(22)

Inequality (18) easily follows after using (21) and (22) in (20), and then taking \(\lambda \rightarrow 0\) while also using the maximum-modulus principle. \(\square \)

Theorem 4

Take \(\mathcal {L}\in \mathcal{S}\mathcal{P}\). Let \(0<C_1\le 1\), \(0<C_2\le 2C_1\) and \(C_3\ge 1\). Let \(s=\sigma +\textrm{i}t\) and

$$\begin{aligned} |t|\ge t_0\ge T_1\ge \max \left\{ \exp {\left( e^{2C_2}\right) },e^{4m/\textrm{d}{}_{\mathcal {L}}},C_3,\exp {\left( e^2\right) }\right\} \end{aligned}$$

with m from axiom (4), such that

$$\begin{aligned}{} & {} t_0-C_3\log {\log {\left( ct_0\right) }}-\frac{1}{2}\ge T_2\ge \max \left\{ T+1,\exp {\left( e^2\right) }\right\} , \end{aligned}$$
(23)
$$\begin{aligned}{} & {} T_1-2C_3\log {\log {T_1}}\ge 0, \end{aligned}$$
(24)

where T and c are as in Theorem 1. Assume that \(\mathcal {L}(z)\ne 0\) for \(\Re \{z\}>1/2\) and \(\left| \Im \{z\}-t\right| \le C_3\log {\log {(c|t|)}}+2\). Then, (4) is true for

$$\begin{aligned} \sigma \in \mathscr {S}\left( C_2,C_2,c,t\right) , \end{aligned}$$
(25)

where \(\mathscr {S}\) is defined by (6),

$$\begin{aligned}{} & {} a_1 \mathrel {\mathop :}=\frac{m}{C_2}\exp {\left( \left( 1+\frac{\log ^{+}{b_1}}{\log {\log {T_1}}}\right) \mathcal {R}_1\right) }, \nonumber \\{} & {} b_1 = b_1\left( \textrm{d}{}_{\mathcal {L}},m,C,C_1,C_3,T_1,T_2\right) \nonumber \\{} & {} \quad \mathrel {\mathop :}=\frac{1}{m}K\left( \textrm{d}{}_{\mathcal {L}},m,C,C_1,T_2\right) \left( 1+\frac{\log {\left( 1+\frac{C_3\log {\log {T_1}}+3/2}{T_1}\right) }}{\log {T_1}}\right) \nonumber \\{} & {} \qquad \times \left( 1+\frac{\log {\left( 1+\frac{1}{\log {T_1}}\log {\left( 1+\frac{2C_3\log {\log {T_1}}+1}{T_1}\right) }\right) }}{\log {\log {T_1}}}\right) \end{aligned}$$
(26)

and

$$\begin{aligned} \mathcal {R}_1=\mathcal {R}\left( C_2,C_3,T_1\right) \mathrel {\mathop :}=\left( 2C_2 + \frac{1}{2C_3}\right) \left( 1-\frac{1}{4C_3\log {\log {T_1}}}\right) ^{-1}, \end{aligned}$$
(27)

while K is defined by (19) and C is from inequality (2).

Proof

We can assume that \(\mathcal {L}\not \equiv 1\) since otherwise the result is trivial. Let \(\delta _0\mathrel {\mathop :}=C_2/\log {\log {(c|t|)}}\) and \(\sigma _0\mathrel {\mathop :}=C_3\log {\log {(c|t|)}} + 1 + \delta _0\), and define also

$$\begin{aligned} \mathscr {D}_1&\mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\left| \sigma _0+\textrm{i}t-z\right| \le \sigma _0-1-\delta _0\right\} , \\ \mathscr {D}_2&\mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\left| \sigma _0+\textrm{i}t-z\right| \le \sigma _0-\sigma \right\} , \\ \mathscr {D}_3&\mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\left| \sigma _0+\textrm{i}t-z\right| \le \sigma _0-1/2-\delta _0\right\} . \end{aligned}$$

Observe that \(\mathscr {D}_j\) are closed discs with the same centre, and

$$\begin{aligned} \mathscr {D}_1 \subseteq \mathscr {D}_2 \subseteq \mathscr {D}_3 \subset \left\{ z\in \mathbb {C}:\Re \{z\}>\frac{1}{2}, \left| \Im \{z\}-t\right| <C_3\log {\log {(c|t|)}}+2\right\} . \end{aligned}$$

Since \(\log {\mathcal {L}(z)}\) is a holomorphic function on the last domain, Theorem 7 implies \(M_2 \le M_1^{1-\mu }M_3^{\mu }\), where

$$\begin{aligned} M_j\mathrel {\mathop :}=\max _{z\in \partial \mathscr {D}_j}\left\{ \left| \log {\mathcal {L}(z)}\right| \right\} , \quad \mu \mathrel {\mathop :}=\left( \log {\frac{\sigma _0-\sigma }{\sigma _0-1-\delta _0}}\right) \left( \log {\frac{\sigma _0-1/2-\delta _0}{\sigma _0-1-\delta _0}}\right) ^{-1}. \end{aligned}$$

Note that \(\left| \log {\mathcal {L}\left( s\right) }\right| \le M_2\) since \(s\in \mathscr {D}_2\).

We need to estimate \(M_1\) and \(M_3\). By (11), we have

$$\begin{aligned} M_1 \le \sup _{\sigma \ge 1+\delta _0}\left\{ \left| \log {\mathcal {L}(s)}\right| \right\} \le \frac{m}{\delta _0}. \end{aligned}$$

We are using Lemma 1 in order to estimate \(M_3\). Let

$$\begin{aligned}{} & {} \mathscr {S}_1 \mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\frac{1}{2}+\delta _0 \le \Re \{z\}\le \frac{3}{2}, \left| \Im \{z\}-t\right| \le \sigma _0-\frac{1}{2}-\delta _0\right\} , \\{} & {} \mathscr {S}_2 \mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\Re \{z\}\ge \frac{3}{2}, \left| \Im \{z\}-t\right| \le \sigma _0-\frac{1}{2}-\delta _0\right\} . \end{aligned}$$

Observe that \(\mathscr {D}_3\subseteq \mathscr {S}_1\cup \mathscr {S}_2\). Inequality (10) implies

$$\begin{aligned} \sup _{z\in \mathscr {S}_2}\left\{ \left| \log {\mathcal {L}(z)}\right| \right\} \le m\log {\zeta \left( \frac{3}{2}\right) } < m. \end{aligned}$$
(28)

Remember that \(c\ge 1\). Take

$$\begin{aligned} \delta (t'){} & {} \mathrel {\mathop :}=\frac{C_2}{\log {\log {\left( c\left( |t'|+\sigma _0-1/2-\delta _0\right) \right) }}},\\{} & {} \quad t'\in \left[ t-\sigma _0+\frac{1}{2}+\delta _0,t+\sigma _0-\frac{1}{2}-\delta _0\right] . \end{aligned}$$

Since \(2C_1-C_2\ge 0\), we have

$$\begin{aligned} \mathscr {S}_1 \subseteq \bigcup _{t':\left| t'-t\right| \le \sigma _0-\frac{1}{2}-\delta _0} \mathscr {D}\left( C_1,\delta (t'),t'\right) , \end{aligned}$$

where the closed disc \(\mathscr {D}\left( C_1,\delta (t'),t'\right) \) is defined by (17). Since

$$\begin{aligned} |t|-C_3\log {\log {(c|t|)}}-\frac{1}{2}\le |t'|\le |t|+C_3\log {\log {(c|t|)}}+\frac{1}{2} \end{aligned}$$

and \(|t|-C_3\log {\log {(c|t|)}}-1/2\) is an increasing function in |t| since \(|t|\log {|t|}\ge C_3\), we have

$$\begin{aligned} |t'|\ge t_0'\mathrel {\mathop :}=t_0-C_3\log {\log {\left( ct_0\right) }}-\frac{1}{2}\ge T_2\ge \max \left\{ T+1,\exp {\left( e^2\right) }\right\} \end{aligned}$$

due to (23). Also, \(0< \delta (t')\le 1/2\) since \(|t'|+\sigma _0-1/2-\delta _0\ge |t|\) and \(\log {\log {(c|t|)}}\ge 2C_2>0\). Furthermore, \(\mathcal {L}(z)\ne 0\) for \(\Re \{z\}>1/2\) and \(\left| \Im \{z\}-t'\right| \le 1\). Conditions of Lemma 1 are thus satisfied, therefore

$$\begin{aligned}{} & {} \sup _{z\in \mathscr {S}_1}\left\{ \left| \log {\mathcal {L}(z)}\right| \right\} \le \frac{1}{C_2}K\left( \text {d}{}_{\mathcal {L}},m,C,C_1,T_2\right) \nonumber \\{} & {} \quad \times \log {\log {\left( c\left( |t|+2C_3\log {\log {(c|t|)}}+1\right) \right) }}\nonumber \\ {}{} & {} \quad \times \log {\left( c\left( |t|+C_3\log {\log {(c|t|)}}+\frac{3}{2}\right) \right) }. \end{aligned}$$
(29)

Note that \(K\ge \textrm{d}{}_{\mathcal {L}}/4\) and \(\log {(c|t|)}\ge 4\,m/\textrm{d}{}_{\mathcal {L}}\). This implies that the right-hand side of (29) is always greater than m, which, together with (28), guarantees that

$$\begin{aligned} \frac{\delta _0 M_3}{m} \le b_1\log {(c|t|)}, \end{aligned}$$
(30)

where \(b_1\) is defined by (26). Here, we also used the fact that

$$\begin{aligned} \frac{1}{\log {(c|t|)}}\log {\left( 1+\frac{\alpha _1C_3\log {\log {(c|t|)}}+\alpha _2}{|t|}\right) } \end{aligned}$$

is a decreasing function in \(c\ge 1\) for \(0<\alpha _1\le 2\) and \(\alpha _2>0\). By using \(\log {(1+u)}\ge u\log {2}\) for \(u\in [0,1]\), simple derivative analysis shows that this is true because

$$\begin{aligned}{} & {} \left( |t|+\alpha _2+\alpha _1 C_3\log {\log {(c|t|)}}\right) \log {\left( 1+\frac{\alpha _1 C_3\log {\log {(c|t|)}}+\alpha _2}{|t|}\right) } \\{} & {} \quad \ge |t|\log {\left( 1+\frac{\alpha _1 C_3\log {\log {|t|}}}{|t|}\right) } \ge \alpha _1\left( \log {2}\right) C_3\log {\log {|t|}} \ge \alpha _1C_3 \end{aligned}$$

since

$$\begin{aligned} \frac{\alpha _1 C_3\log {\log {|t|}}}{|t|} \le \frac{2C_3\log {\log {T_1}}}{T_1} \le 1 \end{aligned}$$

by (24). Furthermore, observe that \(b_1\ge \textrm{d}{}_{\mathcal {L}}/(4\,m)\).

Writing \(\log {(1+u)}=u+R_1(u)\), where \(u\ge 0\) and \(\left| R_1(u)\right| \le u^2/2\), one can easily deduce that \(\mu = 2(1-\sigma )+R\), where

$$\begin{aligned} R = \frac{2\delta _0 + 2\left( \sigma _0-1-\delta _0\right) \left( R_1\left( \frac{1+\delta _0-\sigma }{\sigma _0-1-\delta _0}\right) -2(1-\sigma )R_1\left( \frac{1}{2\left( \sigma _0-1-\delta _0\right) }\right) \right) }{1+2\left( \sigma _0-1-\delta _0\right) R_1\left( \frac{1}{2\left( \sigma _0-1-\delta _0\right) }\right) }. \end{aligned}$$

Since

$$\begin{aligned}{} & {} 1-2\left( \sigma _0-1-\delta _0\right) \left| R_1\left( \frac{1}{2\left( \sigma _0-1-\delta _0\right) }\right) \right| \ge 1-\frac{1}{4C_3\log {\log {T_1}}} > 0, \\{} & {} \quad -\frac{1}{2} \le -\delta _0 \le 1-\sigma \le \frac{1}{2}-\delta _0 < \frac{1}{2}, \end{aligned}$$

it follows that \(0\le 1+\delta _0-\sigma \le 1/2\) and

$$\begin{aligned} \left| R\right| \le \frac{\mathcal {R}_1}{\log {\log {(c|t|)}}}, \end{aligned}$$
(31)

where \(\mathcal {R}_1\) is defined by (27).

We are now in the position to estimate \(M_2\). Since \(0<\mu \le 1\), we now have

$$\begin{aligned} M_2 \le \frac{m}{\delta _0}\left( \frac{\delta _0 M_3}{m}\right) ^{\mu } \le \frac{m}{C_2}\left( b_1\log {(c|t|)}\right) ^{R}\left( b_1\log {(c|t|})\right) ^{2(1-\sigma )}\log {\log {(c|t|)}} \end{aligned}$$

by inequality (30). Since \(b_1\log {(c|t|)}\ge 1\), this and (31) then imply

$$\begin{aligned} \left( b_1\log {(c|t|)}\right) ^{R} \le \left( b_1\log {(c|t|)}\right) ^{\frac{\mathcal {R}_1}{\log {\log {(c|t|)}}}} = \exp {\left( \left( 1+\frac{\log {b_1}}{\log {\log {(c|t|)}}}\right) \mathcal {R}_1\right) }. \end{aligned}$$

The proof of Theorem 4 is thus complete. \(\square \)

Corollary 4

Take \(\mathcal {L}\in \mathcal{S}\mathcal{P}\). Let \(0<C_1\le 1\), \(0<C_2\le 2C_1\), \(C_3\ge 1\) and \(0<C_4\le C_2/2.0001\). Let \(s=\sigma +\textrm{i}t\) and

$$\begin{aligned} |t|\ge t_0\ge T_1\ge \max \left\{ \exp {\left( e^{2\left( 1.00006C_2+C_4\right) }\right) },e^{4m/\textrm{d}{}_{\mathcal {L}}},C_3,\exp {\left( e^2\right) }\right\} +1 \end{aligned}$$
(32)

with m from axiom (4), such that

$$\begin{aligned}{} & {} t_0-C_3\log {\log {\left( ct_0\right) }}-\frac{3}{2}\ge T_2\ge \max \left\{ T+1,\exp {\left( e^2\right) }\right\} , \end{aligned}$$
(33)
$$\begin{aligned}{} & {} T_1-2C_3\log {\log {T_1}}\ge 1, \end{aligned}$$
(34)

where T and c are as in Theorem 1. Assume that \(\mathcal {L}(z)\ne 0\) for \(\Re \{z\}>1/2\) and \(\left| \Im \{z\}-t\right| \le C_3\log {\log {(c(|t|+1))}}+3\). Then, (5) is true for

$$\begin{aligned} \sigma \in \mathscr {S}\left( 1.00006C_2+C_4,C_4,c,t\right) , \end{aligned}$$
(35)

where \(\mathscr {S}\) is defined by (6),

$$\begin{aligned}{} & {} a_2 \mathrel {\mathop :}=\frac{1.0002m}{C_2C_4}\exp {\left( 2C_4\left( 1+\frac{\log ^{+}{b_2}}{\log {\log {T_1}}}\right) +\left( 1+\frac{\log ^{+}{b_2}}{\log {\log {\left( T_1-1\right) }}}\right) \mathcal {R}_2\right) }, \nonumber \\{} & {} b_2 \mathrel {\mathop :}=b_1\left( \textrm{d}{}_{\mathcal {L}},m,C,C_1,C_3,T_1-1,T_2\right) , \quad \mathcal {R}_2=\mathcal {R}\left( C_2,C_3,T_1-1\right) , \end{aligned}$$
(36)

while \(b_1\) and \(\mathcal {R}\) are defined by (26) and (27), respectively.

Proof

We can assume that \(\mathcal {L}\not \equiv 1\) since otherwise the result is trivial. Let \(\delta \mathrel {\mathop :}=C_4/\log {\log {(c|t|)}}\). Then, \(\delta \in (0,1)\). Observe that

$$\begin{aligned} \left\{ z\in \mathbb {C}:|z-s|\le \delta \right\} \subset \left\{ z\in \mathbb {C}:\Re \{z\}>\frac{1}{2}, \left| \Im \{z\}-t\right| <2\right\} . \end{aligned}$$

Since \(\log {\mathcal {L}(z)}\) is a holomorphic function on the latter domain, we can write

$$\begin{aligned} \frac{\mathcal {L}'}{\mathcal {L}}(s) = \frac{1}{2\pi \textrm{i}}\int _{|z-s|=\delta }\frac{\log {\mathcal {L}(z)}}{(z-s)^2}\textrm{d}{z}, \end{aligned}$$

which implies

$$\begin{aligned} \left| \frac{\mathcal {L}'}{\mathcal {L}}(s)\right| \le \frac{1}{\delta }\max _{|z-s|=\delta }\left\{ \left| \log {\mathcal {L}(z)}\right| \right\} \le \frac{1}{C_4}\max _{z\in \mathscr {K}}\left\{ \left| \log {\mathcal {L}(z)}\right| \right\} \log {\log {(c|t|)}}, \end{aligned}$$
(37)

where \(\mathscr {K}\mathrel {\mathop :}=\left\{ z\in \mathbb {C}:\left| \Re \{z\}-\sigma \right| \le \delta , \left| \Im \{z\}-t\right| \le \delta \right\} \). We are going to use Theorem 4 for \(\sigma =\Re \{z\}\) and \(t=\Im \{z\}\) while \(z\in \mathscr {K}\) in order to estimate the right-hand side of (37).

Take \(z\in \mathscr {K}\). Since \(|t|-1\le \left| \Im \{z\}\right| \le |t|+1\) and \(|t|\ge \exp {\left( e^2\right) }\), we have

$$\begin{aligned} 0.99995 \le \frac{\log {\log {\left( |t|-1\right) }}}{\log {\log {|t|}}} \le \frac{\log {\log {\left( c\left| \Im \{z\}\right| \right) }}}{\log {\log {(c|t|)}}}\le \frac{\log {\log {\left( |t|+1\right) }}}{\log {\log {|t|}}} \le 1.00005. \end{aligned}$$
(38)

Then, (38), together with (35), implies that

$$\begin{aligned}{} & {} \frac{1}{2}+\frac{C_2}{\log {\log {\left( c\left| \Im \{z\}\right| \right) }}} \le \frac{1}{2}+\frac{1.00006C_2}{\log {\log {(c|t|)}}} \le \Re \{z\} \\{} & {} \quad \le 1+\frac{2C_4}{\log {\log {(c|t|)}}} \le 1+\frac{C_2}{\log {\log {\left( c\left| \Im \{z\}\right| \right) }}}. \end{aligned}$$

This confirms the validity of (25). Replace \(T_1\) and \(t_0\) in Theorem 4 with \(T_1'\) and \(t_0'\), where \(T_1'\mathrel {\mathop :}=T_1-1\), \(t_0'\mathrel {\mathop :}=t_0-1\), and \(T_1\) and \(t_0\) are as in Corollary 4. Since \(t_0'\le \left| \Im \{z\}\right| \), inequalities (32), (33) and (34) guarantee the conditions of Theorem 4 on \(T_1'\) and \(t_0'\) are satisfied. Also,

$$\begin{aligned}{} & {} \bigcup _{z\in \mathscr {K}}\left\{ w\in \mathbb {C}:\Re \{w\}>\frac{1}{2},\left| \Im \{w\}-\Im \{z\}\right| \le C_3\log {\log {\left( c\left| \Im \{z\}\right| \right) }}+2\right\} \\{} & {} \quad \subset \left\{ w\in \mathbb {C}:\Re \{w\}>\frac{1}{2},\left| \Im \{w\}-t\right| \le C_3\log {\log {(c(|t|+1))}}+3\right\} \end{aligned}$$

and the latter set is free of zeros of \(\mathcal {L}(s)\) by the assumption. Therefore, all conditions of Theorem 4 are satisfied, thus

$$\begin{aligned}{} & {} \left| \log {\mathcal {L}\left( z\right) }\right| \le \frac{m}{C_2}\exp {\left( \left( 1+\frac{\log ^{+}{b_2}}{\log {\log {\left( T_1-1\right) }}}\right) \mathcal {R}_2\right) } \nonumber \\{} & {} \quad \times \left( b_2\log {\left( c\left| \Im \{z\}\right| \right) }\right) ^{2(1-\Re \{z\})}\log {\log {\left( c\left| \Im \{z\}\right| \right) }}, \end{aligned}$$
(39)

where \(b_2\) and \(\mathcal {R}_2\) are defined as (36). Since

$$\begin{aligned} 2(1-\Re \{z\})\le 2(1-\sigma )+\frac{2C_4}{\log {\log {(c|t|)}}} \le 1-\frac{2.00012C_2}{\log {\log {(c|t|)}}} < 1 \end{aligned}$$

with the second expression being non-negative, and \(b_2\log {\left( c\left| \Im \{z\}\right| \right) }\ge 1\), we have

$$\begin{aligned}{} & {} \left( b_2\log {\left( c\left| \Im \{z\}\right| \right) }\right) ^{2(1-\Re \{z\})} \\{} & {} \quad \le 1.00009\exp {\left( 2C_4\left( 1+\frac{\log ^{+}{b_2}}{\log {\log {T_1}}}\right) \right) }\left( b_2\log {(c|t|)}\right) ^{2(1-\sigma )}. \end{aligned}$$

Using the above inequality and (38) in (39), which is then used in (37), gives the final estimate from Corollary 4. \(\square \)

Proof of Theorem 1

We already proved the first part of the theorem, see Remark 1. The second part is essentially the content of Theorem 4 and Corollary 4. \(\square \)

Proof of Corollary 1

By Example 1, we have \(\textrm{d}{}_\mathcal {L}=m=C=c=1\) and \(T=50\). Take \(t_0=T_1=10^4\), \(C_3=10^3\) and \(T_2=7778\) in Theorem 4 and Corollary 4. With these parameters, conditions (23), (24), (33) and (34) are satisfied. We are optimizing \(C_1\), \(C_2\) and \(C_4\) in order to get the smallest possible values for \(a_1\) and \(a_2\), separately in each of the cases (a) and (b). We obtain the following values:

  1. (a)

    \(C_1=0.25\), \(C_2=0.5\),

  2. (b)

    \(C_1=0.34\), \(C_2=0.67\), \(C_4=0.67/2.0001\).

It is easy to verify that all other conditions of Theorem 4 and Corollary 4 are satisfied with such choice of parameters, and the values from Corollary 1 follow immediately. \(\square \)

Proof of Corollary 2

By Example 2, we have \(\textrm{d}{}_\mathcal {L}=m=C=1\), \(c=q\) and \(T=7778\). Take \(t_0=t_0(q)=10450+10^3\log {\log {q}}\), \(T_1=10^4\), \(C_3=10^3\) and \(T_2=7788\) in Theorem 4 and Corollary 4. It is not hard to see that conditions (23), (24), (33) and (34) are then satisfied for every \(q\ge 2\) since

$$\begin{aligned} t_0(q)-10^3\log {\log {\left( t_0(q)q\right) }}-\frac{3}{2} \end{aligned}$$

is an increasing function in \(q\ge 2\). Taking similar approach as in the proof of Corollary 1, we obtain the same values for the parameters \(C_1\), \(C_2\) and \(C_4\). With this, all other conditions of Theorem 4 and Corollary 4 are also satisfied, and the values from Corollary 2 follow immediately. \(\square \)

Proof of Corollary 3

By Example 3, we have \(\textrm{d}{}_\mathcal {L}=m=n_{\mathbb {K}}\), \(c=5.545\left| \Delta _{n_{\mathbb {K}}}\right| ^{1/n_{\mathbb {K}}}\), \(C=1\) and \(T=7778\). Note that \(c\ge 5.545\). Take \(t_0=9650+10^3\log {\log {c}}\), \(T_1=10188\), \(C_3=10^3\) and \(T_2=7794\) in Theorem 4 and Corollary 4. As in the proof of Corollary 2, it is not hard to see that conditions (23), (24), (33) and (34) are then satisfied. Taking similar approach as in the proof of Corollary 1, we obtain the same values for the parameters \(C_1\), \(C_2\) and \(C_4\). With this all other conditions of Theorem 4 and Corollary 4 are also satisfied, and the values from Corollary 3 follow immediately. \(\square \)

Remark 2

It is easy to find (unconditional) bounds for \(\left| \log {\mathcal {L}(s)}\right| \) and \(\left| \mathcal {L}'(s)/\mathcal {L}(s)\right| \) if \(\sigma \ge 1+B/\log {\log {(c|t|)}}\), where \(c\ge 1\), \(B>0\) and \(|t|\ge t_0>e\): if \(\mathcal {L}\in \mathcal{S}\mathcal{P}\), then

$$\begin{aligned}{} & {} \left| \log {\mathcal {L}(s)}\right| \le m\log {\log {\log {(c|t|)}}} + m\log {\frac{1}{B}} + \frac{m\gamma B}{\log {\log {t_0}}}, \\{} & {} \quad \left| \frac{\mathcal {L}'}{\mathcal {L}}(s)\right| \le m\sum _{p}\sum _{k=1}^{\infty }\frac{\log {p}}{p^{k\sigma }} = -m\frac{\zeta '}{\zeta }\left( \sigma \right) \le \frac{m}{\sigma -1}\le \frac{m}{B}\log {\log {(c|t|)}} \end{aligned}$$

by (12) and [10].

4 Proof of Theorem 2

Before proceeding to the proof of Theorem 2, we will provide a general bound for the Mertens function which is a consequence of Theorem 4 for the Riemann zeta-function. We are using the approach outlined in [30, Remark 1].

Theorem 5

Assume the Riemann hypothesis. Let \(0<C_1\le 1\), \(0<C_2\le 2C_1\) and \(C_3\ge 1\). Let

$$\begin{aligned}{} & {} \frac{1}{2} + \frac{C_2}{\log {\log {T_1}}} \le \sigma _0 < 1, \\{} & {} \quad T_1\ge \max \left\{ \exp {\left( e^{2C_2}\right) },\exp {\left( e^{2}\right) },C_3,\exp {\left( \exp {\left( \frac{1}{2\sigma _0-1}\right) }\right) }\right\} ,\\{} & {} \quad T_1-C_3\log {\log {T_1}}-\frac{1}{2}\ge T_2\ge \exp {\left( e^2\right) }, \quad T_1-2C_3\log {\log {T_1}}\ge 0. \end{aligned}$$

Define

$$\begin{aligned} \varepsilon _0 \mathrel {\mathop :}=\frac{1}{C_2}b_{\zeta }^{2\left( 1-\sigma _0\right) }\exp {\left( \left( 1+\frac{\log ^{+}{b_{\zeta }}}{\log {\log {T_1}}}\right) \mathcal {R}\right) } \frac{\log {\log {T_1}}}{\left( \log {T_1}\right) ^{2\sigma _0-1}}, \end{aligned}$$
(40)

where \(\mathcal {R}=\mathcal {R}\left( C_2,C_3,T_1\right) \) is defined by (27) and

$$\begin{aligned} b_{\zeta } = b_{\zeta }\left( C_1,C_3,T_1,T_2\right) \mathrel {\mathop :}=b_1\left( 1,1,1,C_1,C_3,T_1,T_2\right) \end{aligned}$$

with \(b_1\) defined by (26). Take \(\lambda \in \left( 0,T_1\right] \). If \(\varepsilon _0<1\), then

$$\begin{aligned} \left| M(x)\right|\le & {} 1 + \left( \frac{1}{\pi \sigma _0}\left( 1+\frac{\lambda }{T_1}\right) ^{\sigma _0}\int _{0}^{T_1}\frac{\text {d}{u}}{\left| \zeta \left( \sigma _0+\text {i}u\right) \right| }\right) x^{\sigma _0} \nonumber \\ {}{} & {} +\left( 1+\frac{\lambda ^{\varepsilon _0}}{\pi }\left( 1+\frac{\lambda }{T_1}\right) ^{\sigma _0}\right. \nonumber \\ {}{} & {} \times \left. \left( \frac{1}{\varepsilon _0} +\frac{2}{\lambda \left( 1-\varepsilon _0\right) }\left( 1+\frac{\lambda }{T_1}\right) \right) \right) x^{\frac{\sigma _0+\varepsilon _0}{1+\varepsilon _0}} \end{aligned}$$
(41)

for

$$\begin{aligned} x \ge \left( \frac{T_1}{\lambda }\right) ^{\frac{1+\varepsilon _0}{1-\sigma _0}}. \end{aligned}$$
(42)

Proof

Let \(x\ge 1\), \(\widehat{M}(x)\mathrel {\mathop :}=\sum _{n\le x}(x-n)\mu (n)\) and \(0<h\le x\). One can use

$$\begin{aligned} \frac{1}{2\pi \textrm{i}}\int _{c-\textrm{i}\infty }^{c+\textrm{i}\infty } \frac{y^{s}\textrm{d}{s}}{s(s+1)} = \left\{ \begin{array}{ll} 0, &{} 0<y\le 1, \\ 1-y^{-1}, &{} y\ge 1, \end{array} \right. \end{aligned}$$

which is valid for every \(c>0\), to deduce that

$$\begin{aligned} \widehat{M}(x+h) - \widehat{M}(x) = \frac{1}{2\pi \textrm{i}}\int _{\sigma _0-\textrm{i}\infty }^{\sigma _0+\textrm{i}\infty } \frac{(x+h)^{s+1}-x^{s+1}}{s(s+1)\zeta (s)}\textrm{d}{s} \end{aligned}$$
(43)

is true on RH by following the proof of Perron’s formula. Note that

$$\begin{aligned} \left| \left( \widehat{M}(x+h) - \widehat{M}(x)\right) h^{-1} - M(x)\right| \le h+1. \end{aligned}$$
(44)

Let \(\kappa \in (0,1)\) and take \(h=x^{\kappa }\). Assume that \(\lambda xh^{-1}\ge T_1\), which is equivalent to

$$\begin{aligned} x \ge \left( \frac{T_1}{\lambda }\right) ^{\frac{1}{1-\kappa }}. \end{aligned}$$
(45)

The integral in (43) can be written as

$$\begin{aligned}{} & {} \left( \int _{\sigma _0-\textrm{i}T_1}^{\sigma _0+\textrm{i}T_1}+\left( \int _{\sigma _0-\textrm{i}\lambda x/h}^{\sigma _0-\textrm{i}T_1}+\int _{\sigma _0+\textrm{i}T_1}^{\sigma _0+\textrm{i}\lambda x/h}\right) \right. \\{} & {} \quad \left. +\left( \int _{\sigma _0-\textrm{i}\infty }^{\sigma _0-\textrm{i}\lambda x/h}+\int _{\sigma _0+\textrm{i}\lambda x/h}^{\sigma _0+\textrm{i}\infty }\right) \right) \frac{(x+h)^{s+1}-x^{s+1}}{s(s+1)\zeta (s)}\textrm{d}{s}. \end{aligned}$$

Denote by \(\mathcal {I}_1\), \(\mathcal {I}_2\) and \(\mathcal {I}_3\) the above integrals, grouping as indicated and writing in the same order. Then, (43) and (44) imply

$$\begin{aligned} \left| M(x)\right| \le 1 + x^{\kappa } + \frac{1}{2\pi h}\left( \left| \mathcal {I}_1\right| +\left| \mathcal {I}_2\right| +\left| \mathcal {I}_3\right| \right) . \end{aligned}$$
(46)

In the estimation of the first two integrals we are using

$$\begin{aligned} \left| \frac{(x+h)^{s+1}-x^{s+1}}{s+1}\right| \le h\left( x+h\right) ^{\sigma _0} \le \left( 1+\frac{\lambda }{T_1}\right) ^{\sigma _0}hx^{\sigma _0}, \end{aligned}$$

while the last integral is bounded with the help of

$$\begin{aligned} \left| (x+h)^{s+1}-x^{s+1}\right| \le 2\left( x+h\right) ^{\sigma _0+1} \le 2\left( 1+\frac{\lambda }{T_1}\right) ^{\sigma _0+1}x^{\sigma _0+1}. \end{aligned}$$

In deriving both inequalities, we used (45). By Example 1 and Theorem 4 for \(\mathcal {L}(s)=\zeta (s)\), we obtain

$$\begin{aligned} \log {\left| \frac{1}{\zeta \left( \sigma _0+\textrm{i}t\right) }\right| } \le \varepsilon _0\log {t} \end{aligned}$$

for \(t\ge T_1\), where \(\varepsilon _0\) is defined by (40). Note that all conditions of Theorem 4 are satisfied by the assumptions of Theorem 5. In deriving the last inequality, we also used the fact that \(\left( \log {u}\right) ^{1-2\sigma _0}\log {\log {u}}\) is decreasing function for \(u\ge \exp {\left( \exp {\left( 1/\left( 2\sigma _0-1\right) \right) }\right) }\). Then,

$$\begin{aligned}{} & {} \frac{1}{2\pi h}\left( \left| \mathcal {I}_1\right| +\left| \mathcal {I}_2\right| +\left| \mathcal {I}_3\right| \right) \le \frac{1}{\pi \sigma _0}\left( 1+\frac{\lambda }{T_1}\right) ^{\sigma _0}x^{\sigma _0}\int _{0}^{T_1}\frac{\textrm{d}{u}}{\left| \zeta \left( \sigma _0+\textrm{i}u\right) \right| } \nonumber \\{} & {} \quad + \frac{\lambda ^{\varepsilon _0}}{\pi }\left( 1+\frac{\lambda }{T_1}\right) ^{\sigma _0}\left( \frac{1}{\varepsilon _0}+ \frac{2}{\lambda \left( 1-\varepsilon _0\right) }\left( 1+\frac{\lambda }{T_1}\right) \right) x^{\sigma _0+\left( 1-\kappa \right) \varepsilon _0}. \end{aligned}$$
(47)

Comparison between (46) and (47) reveals that the optimal choice for \(\kappa \) is when \(\kappa =\sigma _0+(1-\kappa )\varepsilon _0\), that is when \(\kappa =\left( \sigma _0+\varepsilon _0\right) /\left( 1+\varepsilon _0\right) \). Inequality (45) then gives (42). Taking (47) into (46) then implies (41). \(\square \)

Proof of Theorem 2

First, we will prove (7). We are using Theorem 5. Take \(\sigma _0=0.98\), \(T_1=2.6 \times 10^7\) and \(T_2=T_1-10^3\log {\log {T_1}}-1/2\), together with \(C_1=C_2=1/2\) and \(C_3=10^3\). Then, all conditions of Theorem 5 are satisfied. Values for \(\sigma _0\) and \(T_1\) were obtained by searching for the smallest possible \(T_1\) such that \(\left( \sigma _0+\varepsilon _0\right) /\left( 1+\varepsilon _0\right) \le 0.99\).

By computer (see Remark 3), we calculated that

$$\begin{aligned} \int _{0}^{11520}\frac{\textrm{d}{u}}{\left| \zeta \left( \sigma _0+\textrm{i}u\right) \right| } \le 12951, \end{aligned}$$
(48)

while using Corollary 1 gives

$$\begin{aligned} \int _{11520}^{T_1} \frac{\textrm{d}{u}}{\left| \zeta \left( \sigma _0+\textrm{i}u\right) \right| } \le \int _{11520}^{T_1} u^{\frac{5.44\log {\log {u}}}{\left( \log {u}\right) ^{2\sigma _0-1}}}\textrm{d}{u} \le 5.946 \times 10^{14}. \end{aligned}$$
(49)

Therefore, we can take \(5.95 \times 10^{14}\) as an upper bound for the integral in (41). We choose \(\lambda =2\) in order to make the third term in (41) as small as possible. Then, (7) is true for \(x\ge 10^{711}\). The proof is complete since

$$\begin{aligned} \left| M(x)\right| \le x \le 555.71x^{0.99} + 1.94 \times 10^{14}x^{0.98} \end{aligned}$$

is true for \(1\le x\le 10^{711}\).

For the proof of (8), we are using

$$\begin{aligned} \left| \sum _{n\le x}\frac{\mu (n)}{n^s} - \frac{1}{\zeta (s)}\right| \le \frac{\left| M(x)\right| }{x^\sigma } + |s|\int _{x}^{\infty }\frac{\left| M(u)\right| }{u^{1+\sigma }}\textrm{d}{u}, \end{aligned}$$

valid for \(x\ge 1\) and \(\sigma >1/2\) on RH. Estimate (8) now follows by taking \(s=1\) in the last inequality while bounding the Mertens function with (7). \(\square \)

Remark 3

Computation (48) was done on Gadi, an HPC cluster at NCI Australia, using 192 cores of Intel Xeon Cascade Lake processors. The integral was approximated by Romberg’s method on intervals of length 10 by using SciPy function scipy.integrate.romberg. It is expected that (49) should be of the order \(10^7\), thus improving the second term in (7). However, the author has found such computations very time consuming when pushing them even only to \(10^5\).