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Integral Representation for Riemann-Siegel $Z(t)$ function

Juan Arias de Reyna

Abstract

We apply Poisson formula for a strip to give a representation of $Z(t)$ by means of an integral. \[F(t)=\int_{-\infty}^\infty \frac{h(x)ζ(4+ix)}{7\coshπ\frac{x-t}{7}}\,dx, \qquad Z(t)=\frac{\Re F(t)}{(\frac14+t^2)^{\frac12}(\frac{25}{4}+t^2)^{\frac12}}.\] After that we get the estimate \[Z(t)=\Bigl(\frac{t}{2π}\Bigr)^{\frac74}\Re\bigl\{e^{i\vartheta(t)}H(t)\bigr\}+O(t^{-3/4}),\] with \[H(t)=\int_{-\infty}^\infty\Bigl(\frac{t}{2π}\Bigr)^{ix/2}\frac{ζ(4+it+ix)}{7\cosh(πx/7)}\,dx=\Bigl(\frac{t}{2π}\Bigr)^{-\frac74}\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{2}{1+(\frac{t}{2πn^2})^{-7/2}}.\] We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.

Integral Representation for Riemann-Siegel $Z(t)$ function

Abstract

We apply Poisson formula for a strip to give a representation of by means of an integral. After that we get the estimate with We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.

Paper Structure

This paper contains 5 sections, 22 theorems, 112 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Integral representation for Z(t).
  3. The function f(4+ix).
  4. First approximation to Z(t).
  5. Second approximation to Z(t).

Key Result

Theorem \oldthetheorem

The solution $u(\sigma,t)$ of the Dirichlet problem on the strip $S(a,b)=\{s\in\mathbf{C}\colon a\le \sigma\le b\}$ with boundary values $u(a,t)=A(t)$ and $u(b,t)=B(t)$, where $A(t)$ and $B(t)=\mathop{\hbox{O}}\nolimits(e^{k|t|})$ with $0<k<\pi/(b-a)$ is given by where

Figures (1)

  • Figure 1: x-ray of $H(z)$ for $z\in(10000,10020)\times(-2,4)$

Theorems & Definitions (44)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 34 more