Dirichlet $L$-functions on the critical line and multiplicative chaos
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909866754310144 |
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| author | Vihko, Sami |
| author_facet | Vihko, Sami |
| contents | In this paper we prove that the Dirichlet $L$-functions $L(1/2+ix,χ_q)$, where $χ_q$ is uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converges to a random Schwartz distribution $ζ_{\mathrm{rand}}$, which is related to (complex) Gaussian multiplicative chaos. This is the same limiting object that appeared in [34], where the authors proved that the random shifts of the Riemann zeta function on the critical line $ζ(1/2+ix+iωT)$, where $ω\sim \mathrm{Unif} ([0,1])$, converge as $T\to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dirichlet $L$-functions on the critical line and multiplicative chaos Vihko, Sami Number Theory Probability 60G20, 60F17, 11M06 In this paper we prove that the Dirichlet $L$-functions $L(1/2+ix,χ_q)$, where $χ_q$ is uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converges to a random Schwartz distribution $ζ_{\mathrm{rand}}$, which is related to (complex) Gaussian multiplicative chaos. This is the same limiting object that appeared in [34], where the authors proved that the random shifts of the Riemann zeta function on the critical line $ζ(1/2+ix+iωT)$, where $ω\sim \mathrm{Unif} ([0,1])$, converge as $T\to \infty$. |
| title | Dirichlet $L$-functions on the critical line and multiplicative chaos |
| topic | Number Theory Probability 60G20, 60F17, 11M06 |
| url | https://arxiv.org/abs/2506.16115 |