Simultaneous non-vanishing of Dirichlet L-functions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917404344320000 |
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| author | Bui, Hung M. Florea, Alexandra Milinovich, Micah B. |
| author_facet | Bui, Hung M. Florea, Alexandra Milinovich, Micah B. |
| contents | In this paper, we prove the simultaneous non-vanishing of four Dirichlet $L$-functions at any point on the critical line. More precisely, let $χ_1,\ldots,χ_4$ be even Dirichlet characters modulo $D_1,\ldots, D_4$ respectively, where the $D_j$ are pairwise co-prime and square-free integers. Under the Generalized Riemann Hypothesis, we prove that $\prod_{j=1}^4 L(1/2+it,χχ_j) \neq 0$ for a positive proportion of Dirichlet characters $χ\pmod q$, with $q$ prime and sufficiently large in terms of the $D_j$ and $t$ (and with an explicit relationship between $D_j, t$ and $q$). Unconditionally, we also prove a simultaneous non-vanishing result for four Dirichlet $L$-functions for infinitely many characters $χ\pmod q$, though in this case the proportion tends to zero as $q \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simultaneous non-vanishing of Dirichlet L-functions Bui, Hung M. Florea, Alexandra Milinovich, Micah B. Number Theory In this paper, we prove the simultaneous non-vanishing of four Dirichlet $L$-functions at any point on the critical line. More precisely, let $χ_1,\ldots,χ_4$ be even Dirichlet characters modulo $D_1,\ldots, D_4$ respectively, where the $D_j$ are pairwise co-prime and square-free integers. Under the Generalized Riemann Hypothesis, we prove that $\prod_{j=1}^4 L(1/2+it,χχ_j) \neq 0$ for a positive proportion of Dirichlet characters $χ\pmod q$, with $q$ prime and sufficiently large in terms of the $D_j$ and $t$ (and with an explicit relationship between $D_j, t$ and $q$). Unconditionally, we also prove a simultaneous non-vanishing result for four Dirichlet $L$-functions for infinitely many characters $χ\pmod q$, though in this case the proportion tends to zero as $q \to \infty$. |
| title | Simultaneous non-vanishing of Dirichlet L-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.11941 |