Effective Brauer-Siegel theorems for Artin $L$-functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915530966827008 |
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| author | Cho, Peter J. Oliver, Robert J. Lemke Zaman, Asif |
| author_facet | Cho, Peter J. Oliver, Robert J. Lemke Zaman, Asif |
| contents | Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $ζ_K(s)$ and used this to give effective upper and lower bounds on the residue of $ζ_K(s)$ at $s=1$. We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-functions at $s=1$ that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis. Our bounds are completely unconditional, and rely on no unproven hypotheses about Artin $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective Brauer-Siegel theorems for Artin $L$-functions Cho, Peter J. Oliver, Robert J. Lemke Zaman, Asif Number Theory 11M20, 11M41, 11R42 Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $ζ_K(s)$ and used this to give effective upper and lower bounds on the residue of $ζ_K(s)$ at $s=1$. We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-functions at $s=1$ that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis. Our bounds are completely unconditional, and rely on no unproven hypotheses about Artin $L$-functions. |
| title | Effective Brauer-Siegel theorems for Artin $L$-functions |
| topic | Number Theory 11M20, 11M41, 11R42 |
| url | https://arxiv.org/abs/2510.02309 |