Explicit $L^2$ bounds for the Riemann $\zeta$ function
| Autor: | Dona, Daniele, Helfgott, Harald A., Alterman, Sebastian Zuniga |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | J. Th\'eor. Nombres Bordeaux, 34(1):91--133, 2022 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.5802/jtnb.1194 |
| Popis: | Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large $T$, is based on classical lines, starting with an approximation to $\zeta$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<\sigma\leq 1$ and are strong enough to be of practical use for the rigorous computation of improper integrals. We also present bounds for the $L^{2}$ norm of $\zeta$ in $[1,T]$ for $0\leq\sigma\leq 1$. Comment: 37 pages; v7: version accepted by the journal |
| Databáze: | arXiv |
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