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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.00331 |
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Table of Contents:
- 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.