On the Lindelöf Hypothesis for the Riemann Zeta function and Piltz divisor problem
Lahoucine Elaissaoui
TL;DR
The paper analyzes the Lindelöf Hypothesis through a Fourier-Laguerre expansion of $\zeta^k(s)$ in the half-plane $\sigma>\tfrac{1}{2}$, showing that $\zeta^k(s)$ admits a convergent expansion $\zeta^k(s)=\sum_{n \ge -k} (-1)^n \ell_{n,k} \left(\frac{s-1}{s}\right)^n$ and that the Lindelöf Hypothesis is equivalent to the square-summability of $\ell_{n,k}$ (equivalently $h_k \in H^2(\mathbb{D})$). The coefficients $\ell_{n,k}$ are explicitly expressed in terms of the auxiliary quantities $\lambda_{j,k}$ via a binomial transform, enabling growth bounds $\ell_{n,k}=O(n^{(k-2)/2+\varepsilon})$ and a detailed structural link to the Stieltjes constants. The second part extends these ideas to the Piltz divisor problem, linking the error term $\Delta_k(x)$ with Laguerre expansions, Hardy-space properties, Poisson kernels, and Mellin transforms, and providing equivalent $L^2$-type criteria for Lindelöf through Abel summability and related transforms. Collectively, the work offers a novel analytic framework connecting zeta-values in the critical strip, divisor problems, and orthogonal expansions, with implications for verifying or refuting the Lindelöf Hypothesis through harmonic analysis on $\mathbb{D}$ and $\mathbb{R}_+$. The results reveal deep connections between $\Delta_k$, Fourier coefficients of $\zeta^k$, and transform-based representations that could inform future approaches to zeta-growth in the critical strip.
Abstract
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.