On the Lindelöf hypothesis for the Riemann zeta function and Piltz divisor problem

Lahoucine Elaissaoui

Mathematische Nachrichten, 2025, vol. 298, issue 12, 3960-3973

Abstract: In order to well understand the behavior of the Riemann zeta function inside the critical strip, we show, among other things, the Fourier expansion of the ζk(s)$\zeta ^k(s)$ (k∈N$k \in \mathbb {N}$) in the half‐plane ℜs>1/2$\Re s > 1/2$ and we deduce a necessary and sufficient condition for the truth of the Lindelöf hypothesis. Moreover, if Δk$\Delta _k$ denotes the error term in the Piltz divisor problem then for almost all x≥1$x\ge 1$ and any given k∈N$k \in \mathbb {N}$ we have Δk(x)=limρ→1−∑n=0+∞(−1)nℓn,kLnlog(x)ρn$$\begin{equation*} \hspace*{86pt}\Delta _k(x) = \lim _{\rho \rightarrow 1^-}\sum _{n=0}^{+\infty }(-1)^n\ell _{n,k}L_n{\left(\log (x)\right)}\rho ^n \end{equation*}$$where (ℓn,k)n$(\ell _{n,k})_{n}$ and Ln$L_n$ denote, respectively, the Fourier coefficients of ζk(s)$\zeta ^k(s)$ and Laguerre polynomials.

Date: 2025
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