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Lahoucine Elaissaoui

Publications and source records attributed to Lahoucine Elaissaoui.

7 recordsLinked to original sources
arXiv2025

On the Lindelöf Hypothesis for the Riemann Zeta function and Piltz divisor problem

In order to well understand the behaviour of the Riemann zeta function inside the critical strip, we show; among other things, the Fourier expansion of the $ζ^k(s)$ ($k \in \mathbb{N}$) in the half-plane $\Re s > 1/2$ and we deduce a necessary and sufficient condition for the truth of the Lindelöf Hypothesis. Moreover, if $Δ_k$denotes the error term in the Piltz divisor problem then for almost all $x\geq 1$ and any given $k \in \mathbb{N}$ we have $$Δ_k(x) = \lim_{ρ\to 1^-}\sum_{n=0}^{+\infty}(-1)^n\ell_{n,k}L_n\left(\log(x)\right)ρ^n $$ where $(\ell_{n,k})_{n}$ and $L_n$ denote, respectively, the Fourier coefficients of $ζ^k(s)$ and Laguerre polynomials.

math.NT↗
arXiv2021

On expansions involving the Riemann zeta function and its derivatives

By studying the spectral aspects of the fractional part function in a well-known separable Hilbert space, we show, among other things, a rational approximation of the Riemann zeta function and its derivatives valid on every vertical line in the right half-planes $\Re s> 1/2$ and $\Re s >0.$ Moreover, we provide some discussions and explicit computations related to the fractional part function.

math.NT↗
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↗
arXiv2017

Evaluation of Log-tangent Integrals by series involving $ζ(2n+1)$

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \fracπ{2} \right]$ may be determined by a finite or infinite sum involving the Riemann Zeta-function at odd positive integers.

math.NT↗