Central non-vanishing of Dirichlet $L$-functions

Benjamin Durkan; Andrew Pearce-Crump

doi:10.48550/arXiv.2609.06167

Central non-vanishing of Dirichlet $L$-functions

Benjamin Durkan
, Andrew Pearce-Crump

Department of Mathematics

Research output: Preprint/Working paper › Preprint

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Abstract

We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $χ$ modulo $q$ have $L(1/2,χ)\ne0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.

Original language	English
Publisher	arXiv
Pages	1-61
Number of pages	61
DOIs	https://doi.org/10.48550/arXiv.2609.06167
Publication status	Submitted - 5 Sept 2026

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math.NT

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10.48550/arXiv.2609.06167Licence: CC BY

central_nonvanishingSubmitted manuscript, 356 KBLicence: CC BY
2609.06167v1

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title = "Central non-vanishing of Dirichlet \$L\$-functions",

abstract = "We prove that, for every sufficiently large modulus \$q\textbackslash{}not\textbackslash{}equiv2\textbackslash{}bmod4\$, at least \$3/8-o(1)\$ of the primitive Dirichlet characters \$χ\$ modulo \$q\$ have \$L(1/2,χ)\textbackslash{}ne0\$. We also establish stronger proportions for moduli \$q\$ satisfying certain arithmetic conditions.",

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AB - We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $χ$ modulo $q$ have $L(1/2,χ)\ne0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.

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Central non-vanishing of Dirichlet $L$-functions

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