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

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.