Fourier Expansion of the Riemann zeta function and applications

TL;DR

This paper investigates the Fourier expansion of the Riemann zeta function in the half-plane and explores its implications for understanding the function's value distribution and related hypotheses.

Contribution

It introduces a Fourier expansion of $\zeta(s)$ in the half-plane with coefficients linked to Stieltjes constants, and applies this to compute related Poisson integrals.

Findings

Fourier expansion of $\zeta(s)$ in $ ext{Re} s \\geq 1/2$

Explicit computation of Poisson integrals for $\zeta(s)$

Discussion on implications for Riemann and Lindel"{o}f hypotheses

Abstract

We study the distribution of values of the Riemann zeta function $\zeta(s)$ on vertical lines $\Re s + i \mathbb{R}$, by using the theory of Hilbert space. We show among other things, that, $\zeta(s)$ has a Fourier expansion in the half-plane $\Re s \geq 1/2$ and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of $\zeta(s) - s/(s-1)$. Moreover, we discuss our results with respect to the Riemann and Lindel\"{o}f hypotheses on the growth of the Fourier coefficients.