Estimates for L-functions in the Critical Strip Under GRH with Effective Applications

Assuming the generalized Riemann hypothesis, we provide explicit upper bounds for moduli of logL(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log {\mathcal {L}(s)}$$\end{document} and L′(s)/L(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}'(s)/\mathcal {L}(s)$$\end{document} in the neighbourhood of the 1-line when L(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}(s)$$\end{document} are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood’s well-known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application, we provide conditional and effective estimates for the Mertens function.


Introduction
Let s = σ + it, where σ and t are real numbers. Determination of the true order of |ζ(s)| 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 ζ (σ + it) ε,σ |t| ε for every ε > 0 and σ ≥ 1/2, which in the case σ = 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 σ = 1/2. For some explicit results in this direction, see [12][13][14]23,24,34,35].
The sets R and 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 (exp(10)), while the author [29, Corollary 1] derived a bound for the Riemann zeta-function which is valid for all t ≥ 2π. Effective upper and lower bounds for ζ(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) = n≤x μ(n) and for the number of k-free numbers, see [30,Theorem 2], where x ≥ 1 and μ(n) is the Möbius function. Unfortunately, the bounds we have obtained are valid only for very large x, for example |M (x)| ≤ 0.51x 0.99 log x, x ≥ 10 10 4. 6 .
Similarly, estimates for m(x) = n≤x μ(n)/n were also provided. A method [30,Remark 1] was proposed to extend validity of estimates of the form M (x) x α for fixed α ∈ (1/2, 1) by employing bounds like (4). Here, we are able to prove the following.  (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 ζ(s), L(s, χ) and ζ 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.

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).

Preliminaries
The Selberg class SP of functions with a polynomial Euler product consists of Dirichlet series satisfying the following axioms: (1) Ramanujan hypothesis. We have a(n) ε n ε for any ε > 0.
(2) Analytic continuation. There exists k ∈ N 0 such that (s − 1) k L(s) is an entire function of finite order. with (Q, λ j ) ∈ R 2 + , and (μ j , ω) ∈ C 2 with {μ j } ≥ 0 and |ω| = 1. (4) Polynomial Euler product. There exists m ∈ N, and for every prime It is well known that axiom (1) implies the absolute convergence of (9) in the half-plane σ > 1, and that axioms (1) and (4) imply that |α j (p)| is true for σ ≥ σ 0 > 1. Inequality (10) implies the following two approximations where γ 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 σ 0 is close to 1. Note that SP ⊆ S where S is the classical Selberg class of functions introduced in [28], i.e. axiom (4) is replaced by and |t| ≤ T , it follows that d L is well defined although parameters from axiom (3) are not unique. Note that d 1 = 0 and d ζ = 1. It is known [32, Theorem 6.1] that d L ≥ 1 for every L ∈ S\{1}, and it is conjectured that d L is always a positive integer. Kaczorowski and Perelli proved that ζ(s) and shifts L(s + iθ, χ), θ ∈ R, of Dirichlet L-functions (σ > 1) attached to a primitive character χ modulo q ≥ 2, are the only functions in S with degree 1, see [31] for a simplified proof. Important examples are also Dedekind zeta-functions (σ > 1) where K is a number field, N (·) is the norm of an ideal, a runs through all non-zero ideals and p runs through all prime ideals of the ring of integers of K. The last equality follows because any rational prime number p has a unique factorization (p) = r j=1 p ej j with N (p j ) = p fj and r j=1 e j f j = n K := [K : Q], where the positive integers e j , f j and r depend on p. Therefore, r ≤ n K , which implies a polynomial Euler product representation for m = n K . We have that ζ K belongs to SP and d ζ K = n K . Observe also that ζ Q (s) = ζ(s).
Taking L ∈ S, we can use Stirling's formula to prove where this estimate is uniform in σ ∈ [σ 1 , σ 2 ] for fixed σ 1 ≤ σ 2 , see [32,Lemma 6.7]. It is possible to make (13) uniform also in L by means of the data of the functional equation, but such an approach is not needed in the present paper.

On the Growth of L(s) on the 1-Line
It is convenient to introduce an additional axiom which concerns the growth of L (1 + it) when L ∈ S and |t| → ∞.
(5) Growth on the 1-line. L (1 + it) log |t| for some > 0. In the case of the Riemann zeta-function, it is a standard result that we can take = 1, while a substantial improvement to = 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 ζ(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 G. We say that L ∈ G, if the series (9) is absolutely convergent for σ > 1, a(1) = 1, and L satisfies beside axioms (2) and (4) also the following two axioms: (6) Zero-free region. There exists c L > 0 such that L has no zeros in the region , except the possible Siegel zero, i.e. real exceptional zero of L in the neighbourhood of 1.
Observe that here class G does not require a functional equation; axiom (3) implies axiom (7), but the latter is sufficient to show that then L(s) is polynomially bounded in vertical strips by using the Phragmén-Lindelöf principle. It is expected that S ⊆ G.
The next theorem explores possible connections between classes G, S, SP and axiom (5). As usual, d α (n) denotes the number of ways positive integer n can be written as a product of α ≥ 2 factors, and we extend this to d 1 (n) ≡ 1. Proof. First, we are going to prove the assertion (a) by following the method from [11]. Take L ∈ G. Let X ≥ 2, σ > 1 and since p ks = p kσ ≥ p k . For t ≥ 2, α > 0 and ε > 0, define By axiom (6) there exist α and t 0 > 0 such that there are no zeros of L(s) in a neighbourhood of C for t ≥ t 0 . Moreover, one can use Theorem 6 together with axiom (7) and inequality (10) to prove that log L(z) log t for z ∈ ∂C .
Take X = exp (log t) 1+ε/m . By (14), (15) and Cauchy's formula, we then We obtain with the same result also for the third integral while the second integral may be bounded as 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 L ∈ S, x ≥ 1 and t 0 > 0 sufficiently large. We can assume that L ≡ 1 since otherwise the result is trivial.
The first equality follows from the classical Mellin integral, while the second follows by moving the line of integration to {z} = −3/2, using the functional equation, and detecting two poles at z = 0 and z = −it 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 I and let z = −3/2 + iu, u ∈ R. Then, while the implied constants are uniform in u and t. Obviously, L (it −z) 1. Splitting the range of integration in I into two parts, |u| ≤ log |t| and |u| > From the last expression, we can see that where we used a(n) n and log 2 |t| + u ≤ u log |t| , the last inequality valid for u ≥ 2 and log |t| ≥ 5. All of these finally imply Since a(n) d α (n) by the assumption, and n≤X d α (n) In correspondence with these functions, our condition a(n) d α (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.

Convexity Estimates for L(s)
Assuming axiom (5), it is easy to prove the precise form of the convexity-type result for L(s). Proposition 1. Take σ 0 < 0, L ∈ S and assume that L also satisfies axiom (5). Then, where the implied constants are uniform in σ. Also, the estimate for σ ∈ [σ 0 , 0) follows from the estimate for σ > 1 and the functional equation, so it remains to prove the bounds for σ ≥ 0.
where k is from axiom (2). Then, f L (s) and g L (s) are holomorphic functions of finite order in the half-plane {z ∈ C : {z} > −1}.
Since |f L (1 + it)| and |f L (it)| are bounded for all t ∈ R, the Phragmén-Lindelöf theorem implies that also |f L (s)| is bounded for σ ∈ [0, 1] and t ∈ R. This proves the first estimate.
Trivially, L(s) log |t| for σ ≥ 2. As before, because |g L (1 + it)| and |g L (2 + it)| are bounded for all t ∈ R, this implies that also |g L (s)| is bounded for all σ ∈ [1,2] and t ∈ R. The proof is thus complete.
We are going to provide numerical values for the constants C, c and T from Theorem 1 in the case when L(s) is ζ(s), L(s, χ), and ζ K (s).

Proof of Theorem 1 and Its Corollaries
In this section, we prove the estimates on log L(s) and L (s)/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 |log L(z)| with the estimate (2) on some particular circles by means of Theorem 7.
m is from axiom (4) and C is from inequality (2).
Proof. Let λ ∈ (0, 1) and define Observe that D = D (C 1 , δ, t ) and D 0 are closed discs with the same centre, and Since log L(z) is a holomorphic function on the last domain, Theorem 6 implies By (2), we have  (21) and (22) in (20), and then taking λ → 0 while also using the maximum-modulus principle.
We are now in the position to estimate M 2 . Since 0 < μ ≤ 1, we now have by inequality (30). Since b 1 log (c|t|) ≥ 1, this and (31) then imply The proof of Theorem 4 is thus complete. with m from axiom (4), such that where T and c are as in Theorem 1. Assume that L(z) = 0 for {z} > 1/2 and | {z} − t| ≤ C 3 log log (c(|t| + 1)) + 3. Then, (5) is true for where S is defined by (6), while b 1 and R are defined by (26) and (27), respectively.
Proof. We can assume that L ≡ 1 since otherwise the result is trivial. Let δ := C 4 / log log (c|t|). Then, δ ∈ (0, 1). Observe that Since log L(z) is a holomorphic function on the latter domain, we can write where K := {z ∈ C : | {z} − σ| ≤ δ, | {z} − t| ≤ δ}. We are going to use Theorem 4 for σ = {z} and t = {z} while z ∈ K in order to estimate the right-hand side of (37). Take z ∈ K . This confirms the validity of (25). Replace T 1 and t 0 in Theorem 4 with T 1 and t 0 , where T 1 := T 1 − 1, t 0 := t 0 − 1, and T 1 and t 0 are as in Corollary 4.
and the latter set is free of zeros of L(s) by the assumption. Therefore, all conditions of Theorem 4 are satisfied, thus where b 2 and R 2 are defined as (36). Since with the second expression being non-negative, and b 2 log (c | {z}|) ≥ 1, we have Using the above inequality and (38) in (39), which is then used in (37), gives the final estimate from Corollary 4. Proof of Corollary 2. By Example 2, we have d 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 ≥ 2 since is an increasing function in q ≥ 2. Taking similar approach as in the proof of Corollary 1, we obtain the same values for the parameters C 1 , C 2 and by (12) and [10].

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].
Proof of Theorem 2. First, we will prove (7). We are using Theorem 5. Take σ 0 = 0.98, T 1 = 2.6 × 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 σ 0 and T 1 were obtained by searching for the smallest possible T 1 such that (σ 0 + ε 0 ) / (1 + ε 0 ) ≤ 0.99. By computer (see Remark 3), we calculated that Therefore, we can take 5.95 × 10 14 as an upper bound for the integral in (41). We choose λ = 2 in order to make the third term in (41) as small as possible. Then, (7) is true for x ≥ 10 711 . The proof is complete since A. Simonič MJOM 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 .
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Appendix
We are stating the Hadamard-Borel-Carathéodory inequality and Hadamard's three-circles theorem.