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arXiv · 1809.02829

Fourier Expansion of the Riemann zeta function and applications

Lahoucine Elaissaoui·Zine El-Abidine Guennoun
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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.

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Lahoucine Elaissaoui, Zine El-Abidine Guennoun. 2018-09-08. Fourier Expansion of the Riemann zeta function and applications. https://doi.org/10.1016/j.jnt.2019.09.025

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