On the Lindelöf 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 Lindelöf Hypothesis. Moreover, if ${\Delta }_k$denotes the error term in the Piltz divisor problem then for almost all $x\ge 1$ and any given $k\in \mathbb{N}$ we have $${\Delta }_k(x)=\underset{\rho \to 1^{-}}{\lim }\sum _{n=0}^{+\infty }(-1{)}^n{\ell }_{n,k}{L}_n\left(\log (x)\right){\rho }^n$$ where $({\ell }_{n,k}{)}_n$ and $L_n$ denote, respectively, the Fourier coefficients of ${\zeta }^k(s)$ and Laguerre polynomials.

18 pages