An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912557888962560 |
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| author | Alberts, Brandon |
| author_facet | Alberts, Brandon |
| contents | Given a Dirichlet series $L(s) = \sum a_n n^{-s}$, the asymptotic growth rate of $\sum_{n\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(σ+it)|$ for fixed real part $σ$. We modify this approach to prove new Tauberian theorems with error terms depending only on the average size of $L(σ+it)$ as $t$ varies, and we take care to track explicit dependence on various parameters. This often leads to stronger error bounds, and introduces strong connections between asymptotic counting problems and moments of $L$-functions.
We provide self-contained statements of Tauberian theorems in anticipation that these results can be used ``out of the box'' to prove new asymptotic expansions. We demonstrate this by proving square root saving error bounds for the number of $C_n$-extensions of $\mathbb{Q}$ of bounded discriminant when $n=3$, $4$, $8$, $16$, or $2p$ for $p$ an odd prime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields Alberts, Brandon Number Theory 11M45, 40E05, 11N37, 11N45, 11R20 Given a Dirichlet series $L(s) = \sum a_n n^{-s}$, the asymptotic growth rate of $\sum_{n\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(σ+it)|$ for fixed real part $σ$. We modify this approach to prove new Tauberian theorems with error terms depending only on the average size of $L(σ+it)$ as $t$ varies, and we take care to track explicit dependence on various parameters. This often leads to stronger error bounds, and introduces strong connections between asymptotic counting problems and moments of $L$-functions. We provide self-contained statements of Tauberian theorems in anticipation that these results can be used ``out of the box'' to prove new asymptotic expansions. We demonstrate this by proving square root saving error bounds for the number of $C_n$-extensions of $\mathbb{Q}$ of bounded discriminant when $n=3$, $4$, $8$, $16$, or $2p$ for $p$ an odd prime. |
| title | An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields |
| topic | Number Theory 11M45, 40E05, 11N37, 11N45, 11R20 |
| url | https://arxiv.org/abs/2508.20814 |