Fourier Expansion of the Riemann zeta function and applications

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 $\Re s\ge 1/2$ 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 Lindelöf hypotheses on the growth of the Fourier coefficients.

21 pages

References in corpus (1)

• Evaluation of some second moment and other integrals for the Riemann, Hurwitz, and Lerch zeta functions

Cited by in corpus (2)

• On expansions involving the Riemann zeta function and its derivatives
• On certain Fourier expansions for the Riemann zeta function