Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros
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Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros ¯ Ramunas Garunkštis1
· Antanas Laurinˇcikas1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract Assume the Riemann hypothesis. On the right-hand side of the critical strip, we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros. Keywords Riemann zeta-function · Riemann hypothesis · Discrete mean square
1 Introduction Let s = σ + it be a complex variable. In this paper, T always tends to plus infinity. Let N (T ) denote the number of zeros of the Riemann zeta-function ζ (s) in the region 0 ≤ σ ≤ 1, 0 < t ≤ T . The Riemann–von Mangoldt formula states (Tichmarsh [23, Theorem 9.4]) that N (T ) =
T T log + O(log T ). 2π 2πe
(1.1)
Let ρ = β + iγ denote a non-real zero of ζ (s). The Riemann hypothesis (RH) states that β = 1/2 for all non-real zeros of the Riemann zeta-function. We prove the following two theorems
The first author is supported by a Grant No. MIP-049/2014 from the Research Council of Lithuania.
B
Ram¯unas Garunkštis [email protected] http://www.mif.vu.lt/∼garunkstis Antanas Laurinˇcikas [email protected]
1
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania
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R. Garunkštis, A. Laurinˇcikas
Theorem 1.1 Assume RH. Let σ > 1/2. Let A > 0 be as large as we like. Then there is δ = δ(A) > 0 such that, for |t| ≤ T A , ζ T T T |ζ (s + iγ )|2 = ζ (2σ ) log + ζ (2σ ) (s + 1/2) + Oσ,A (T 1−δ ). 2π 2πe ζ π 0 1/2. Then, for any ε > 0, |ζ (s + iγ )|2 σ,ε T log T + |t|ε , 0 0 fixed. Instead of arithmetic progressions, Dubickas and Laurinˇcikas [5] considered the set {k α h : k = 0, 1, 2, . . . } with 0 < α < 1 fixed. The related discrete mean square was considered by Gonek [15]. He proved, assuming RH, that for real α, |α| ≤ (1/4π) log(T /2π), 2 1 sin(πα) 2 T 2πα ζ = 1− +i γ + log2 T + O(T log T ). 2 log(T /2π) πα 2π 0 0. Then, for any given positive number δ, there is λ = λ(δ, σ ) > 0 such that 1 ζ (s) = + r (s), ns δ n≤t
where r (s) t −λ . Proof The lemma follows from Tichmarsh [23, Theorem 13.3].
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Discrete mean square of the Riemann zeta-function over…
Lemma 2.2 Assume RH. Let x, T > 1. Then √ T (x) x iγ = − x log(2x T ) log log(3x) √ +O 2π x 0 0. Then (γ + t)−2λ λ T (T + t)−2λ log T λ T (T + t)−λ 0 1/2, t ≥ 0, and 0 < δ < 1. Then there is a positive number λ = λ(δ, σ ) such that ζ T T T |ζ (s + iγ )|2 = ζ (2σ ) log + ζ (2σ ) (s + 1/2) 2π 2πe ζ π γ ≤T
+ O T (T + t)−δ(σ −1/2) + T (T + t)−λ + (T + t)2δ + T 1/2 (T + t)δ−λ . Proof In view of Lemma 2.1, we have
|ζ (σ + iγ + it)|2 =
γ ≤T
+
0
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