On the Lindel\"{o}f Hypothesis for the Riemann Zeta function and Piltz divisor problem

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 $\zeta^k(s)$ ($k \in \mathbb{N}$) in the half-plane $\Re s > 1/2$ and we deduce a necessary and sufficient condition for the truth of the Lindel\"{o}f Hypothesis. Moreover, if $\Delta_k$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 $$\Delta_k(x) = \lim_{\rho \to 1^-}\sum_{n=0}^{+\infty}(-1)^n\ell_{n,k}L_n\left(\log(x)\right)\rho^n $$ where $(\ell_{n,k})_{n}$ and $L_n$ denote, respectively, the Fourier coefficients of $\zeta^k(s)$ and Laguerre polynomials.