arXiv subjects

Zine El-Abidine Guennoun

Publications and source records attributed to Zine El-Abidine Guennoun.

2 recordsLinked to original sources

arXiv2019

Fourier Expansion of the Riemann zeta function and applications

We study the distribution of values of the Riemann zeta function $ζ(s)$ on vertical lines $\Re s + i \mathbb{R}$, by using the theory of Hilbert space. We show among other things, that, $ζ(s)$ 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 $ζ(s) - s/(s-1)$. Moreover, we discuss our results with respect to the Riemann and Lindelöf hypotheses on the growth of the Fourier coefficients.

math.NT↗

arXiv2018

Relating log-tangent integrals with the Riemann zeta function

We show that integrals involving log-tangent function, with respect to certain square-integrable functions on $(0, π/2)$, can be evaluated by some series involving the harmonic number. Then we use this result to establish many closed forms relating to the Riemann zeta function at odd positive integers. In addition, we show that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and that its values depend on the Dirichlet series $ζ_h(s) :=\sum_{n = 1}^\infty h_n n^{-s}$, where $h_n = \sum_{k=1}^n(2k-1)^{-1}$.

math.NT↗