Selberg’s central limit theorem for quadratic dirichlet L-functions over function fields
Selberg’s central limit theorem for quadratic dirichlet L-functions over function fields
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函数域上二次狄利克雷 L 函数的 Selberg 中心极限定理
DOI：
10.1007/s00605-023-01853-y
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Allysa Lumley
中科院分区：
文献类型：
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作者：
P. Darbar;Allysa Lumley
本文研究了在固定有限域$${\mathbb {F}}_q$$ F q上，当极限为$$g\rightarrow \infty $$ g→∞时，与g属超椭圆曲线相关的Dirichlet L -函数辛族中中心值$$L\left( \frac{1}{2}, \chi _D\right) $$ l12， χ D的对数。无条件地，我们证明了$$\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |$$ log | l12， χ D |的分布是由平均值$$\frac{1}{2}\log \deg (D)$$ 1 2 log deg (D)和方差$$\log \deg (D)$$ log deg (D)的全高斯分布渐近有界的，并且$$\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |$$ log | l12， χ D |至少为$$94.27 \%$$ 94.27 % Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.
In this article, we study the logarithm of the central value $$L\left( \frac{1}{2}, \chi _D\right) $$ L 1 2 , χ D in the symplectic family of Dirichlet L -functions associated with the hyperelliptic curve of genus g over a fixed finite field $${\mathbb {F}}_q$$ F q in the limit as $$g\rightarrow \infty $$ g → ∞ . Unconditionally, we show that the distribution of $$\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |$$ log | L 1 2 , χ D | is asymptotically bounded above by the full Gaussian distribution of mean $$\frac{1}{2}\log \deg (D)$$ 1 2 log deg ( D ) and variance $$\log \deg (D)$$ log deg ( D ) , and also $$\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |$$ log | L 1 2 , χ D | is at least $$94.27 \%$$ 94.27 % Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.
超椭圆系综中 $L$ 函数的截断乘积表示
DOI：
10.48550/arxiv.1609.05324
发表时间：
2016
期刊：
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影响因子：
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作者：
Andrade J
通讯作者：
Andrade J