A Central Limit Theorem for Linear Combinations of Logarithms of Dirichlet $L$-functions Sampled at the Zeros of the Zeta Function
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908313742999552 |
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| author | Çiçek, Fatma Gonek, Steven M. Kirila, Scott J. |
| author_facet | Çiçek, Fatma Gonek, Steven M. Kirila, Scott J. |
| contents | Let $L(s, χ_1), \ldots, L(s, χ_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log|L(ρ,χ_1)|+\dots+a_N\log|L(ρ,χ_N)|$ has an approximately normal distribution as $T\to \infty $ with mean $0$ and variance $ \tfrac12 \big({a_1}^2+\dots+{a_N}^2\big)\log\log T.$ Here $a_1, a_2, \ldots, a_N \in \mathbb{R}$, and $ρ$ runs over the nontrivial zeros of the zeta function with $0< \Imρ\leq T$. From this we deduce that the vectors $\big(\log|L(ρ,χ_1)|/\sqrt{ \frac12 \log\log T}, \ldots, \log|L(ρ,χ_N)|/\sqrt{\frac12 \log\log T}\,\big)$ have approximately an $N$-variate normal distribution whose components are approximately mutually independent as $T\to \infty$. We apply these results to study the proportion of the $ρ$ that are zeros or $a$-values of linear combinations of the form $c_1 L(ρ, χ_1)+ \cdots + c_N L(ρ, χ_N)$ with complex $c_i$ as coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Central Limit Theorem for Linear Combinations of Logarithms of Dirichlet $L$-functions Sampled at the Zeros of the Zeta Function Çiçek, Fatma Gonek, Steven M. Kirila, Scott J. Number Theory 11M06, 11M26 Let $L(s, χ_1), \ldots, L(s, χ_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log|L(ρ,χ_1)|+\dots+a_N\log|L(ρ,χ_N)|$ has an approximately normal distribution as $T\to \infty $ with mean $0$ and variance $ \tfrac12 \big({a_1}^2+\dots+{a_N}^2\big)\log\log T.$ Here $a_1, a_2, \ldots, a_N \in \mathbb{R}$, and $ρ$ runs over the nontrivial zeros of the zeta function with $0< \Imρ\leq T$. From this we deduce that the vectors $\big(\log|L(ρ,χ_1)|/\sqrt{ \frac12 \log\log T}, \ldots, \log|L(ρ,χ_N)|/\sqrt{\frac12 \log\log T}\,\big)$ have approximately an $N$-variate normal distribution whose components are approximately mutually independent as $T\to \infty$. We apply these results to study the proportion of the $ρ$ that are zeros or $a$-values of linear combinations of the form $c_1 L(ρ, χ_1)+ \cdots + c_N L(ρ, χ_N)$ with complex $c_i$ as coefficients. |
| title | A Central Limit Theorem for Linear Combinations of Logarithms of Dirichlet $L$-functions Sampled at the Zeros of the Zeta Function |
| topic | Number Theory 11M06, 11M26 |
| url | https://arxiv.org/abs/2504.08322 |