An explicit formula for the square of the Riemann zeta-function on the critical line
type: Item
instance of: scholarly article
title: An explicit formula for the square of the Riemann zeta-function on the critical line
zbMATH Open document ID: 1016.11031
author: A. Miyai
published in: Tsukuba Journal of Mathematics
publication date: 2002-10-31T00:00:00Z
review text: The author proves the explicit formula \[ \begin{aligned} |\zeta(\tfrac 12+ iT)|^2 & = \sqrt{2}\sum_{2T/\pi+1<n\leq T_C(\alpha)}{(-1)^nd(n)\over \sqrt{n}({1\over 4} - T(2\pi n))^{1/4}}\cos(f_C(T,n))\\ & + 2\sum_{n\leq T\alpha/(2\pi)}d(n)n^{-1/2}\cos(T\log(T/(2\pi n e)) - \pi/4) + O(T^{\varepsilon}).\end{aligned} \tag{1} \] Here \(d(n)\) is the number of divisors of \(n\), \(\theta \leq \alpha < 1\) is a constant for a given \(0 < \theta < 1\), \(T_C(\alpha) = (1+\alpha)^2T/(2\pi)\), and \[ f_C(T,n) = 2T\operatorname {arcosh}\left(\sqrt{\pi n\over 2T} \right) - 2\pi n\sqrt{{1\over 4} - {T\over 2\pi n}} + {\pi\over 4}, \] \[ \text{arcosh}(z) = \log(z + \sqrt{z^2-1})\qquad(|z|> 1). \] As an application, by exponential averaging with the Gaussian weight, one obtains from (1) the classical bound \(\zeta({1\over 2} + it) \ll_\varepsilon |t|^{1/6+\varepsilon}\). The proof is similar to the approach used by \textit{M. Jutila} [J. Number Theory 18, 135-156 (1984; Zbl 0533.10034)], who transformed Dirichlet polynomials of the type \(\sum_{N<n\leq N'\leq 2N}d(n)n^{-it}\) with the aid of Voronoi's formula, inserting the (trivial) factor \(1 = \exp(2\pi in)\) in the sum to regulate the location of the saddle points of the ensuing exponential integrals. The present author does not work with Dirichlet polynomials, but works directly with the classical approximate functional equation \[ |\zeta(\tfrac 12+iT)|^2 = 2\sum_{n\leq T/(2\pi)}d(n)n^{-1/2}\cos(T\log(T/(2\pi ne)) - \pi/4) + O(\log T).\tag{2} \] The portion of the sum in (2) for which \(n < \alpha T/(2\pi)\) is kept intact, and the remaining portion is transformed by the Voronoi formula, after the factor \(1 = \exp(-2\pi in)\) is inserted in the sum. This produces the first sum on the right-hand side of (1), plus a small error term. The analysis required for the treatment of various sums and integrals that arise in the course of the proof is delicate, and the author displays considerable skill in dealing with them.
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1822084
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: explicit formula
zbMATH Keywords: saddle point
zbMATH Keywords: Voronoi formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496164286
OpenAlex ID: W326936175
DOI: 10.21099/TKBJM/1496164286
description: scientific article; zbMATH DE number 1822084