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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2019
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| Accès en ligne: | https://arxiv.org/abs/1906.01097 |
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| _version_ | 1866914682031308800 |
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| author | Dona, Daniele Helfgott, Harald A. Alterman, Sebastian Zuniga |
| author_facet | Dona, Daniele Helfgott, Harald A. Alterman, Sebastian Zuniga |
| contents | Explicit bounds on the tails of the zeta function $ζ$ are needed for applications, notably for integrals involving $ζ$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $ζ$.
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 $ζ$ via Euler-Maclaurin.
Both bounds give main terms of the correct order for $0<σ\leq 1$ 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 $ζ$ in $[1,T]$ for $0\leqσ\leq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_01097 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Explicit $L^2$ bounds for the Riemann $ζ$ function Dona, Daniele Helfgott, Harald A. Alterman, Sebastian Zuniga Number Theory 11M06 Explicit bounds on the tails of the zeta function $ζ$ are needed for applications, notably for integrals involving $ζ$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $ζ$. 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 $ζ$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<σ\leq 1$ 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 $ζ$ in $[1,T]$ for $0\leqσ\leq 1$. |
| title | Explicit $L^2$ bounds for the Riemann $ζ$ function |
| topic | Number Theory 11M06 |
| url | https://arxiv.org/abs/1906.01097 |