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 ${\zeta }^k(s)$ ($k\in \mathbb{N}$) in the half-plane $\mathrm{\Re }s>1\mathrm{/}2$ and we deduce a necessary and sufficient condition for the truth of the Lindelöf Hypothesis. Moreover, if ${\mathrm{\Delta }}_k$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 $${\mathrm{\Delta }}_k(x)=\underset{\rho \to 1^{-}}{\lim }\sum _{n=0}^{+\mathrm{\infty }}(-1{)}^n{\mathrm{\ell }}_{n,k}{L}_n(\log (x)){\rho }^n$$ where $({\mathrm{\ell }}_{n,k}{)}_n$ and $L_n$ denote, respectively, the Fourier coefficients of ${\zeta }^k(s)$ and Laguerre polynomials.