Quadratic Euler-Kronecker constants in positive characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917941177483264 |
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| author | Akbary, Amir Boudreau, Félix Baril |
| author_facet | Akbary, Amir Boudreau, Félix Baril |
| contents | In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of a rational global function field, via the values of logarithmic derivatives of Dirichlet $L$-functions at 1. Using a probabilistic model, we show that the values converge to a limiting distribution with a smooth, positive density function, as the genii of quadratic fields approach infinity. We then prove a discrepancy theorem for the convergence of the frequency of these values, and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. Our theorems imply new distribution results on the stable Taguchi heights and logarithmic Weil heights of rank 2 Drinfeld modules with CM. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_00288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic Euler-Kronecker constants in positive characteristic Akbary, Amir Boudreau, Félix Baril Number Theory 11R59, 11R58, 11G09 In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of a rational global function field, via the values of logarithmic derivatives of Dirichlet $L$-functions at 1. Using a probabilistic model, we show that the values converge to a limiting distribution with a smooth, positive density function, as the genii of quadratic fields approach infinity. We then prove a discrepancy theorem for the convergence of the frequency of these values, and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. Our theorems imply new distribution results on the stable Taguchi heights and logarithmic Weil heights of rank 2 Drinfeld modules with CM. |
| title | Quadratic Euler-Kronecker constants in positive characteristic |
| topic | Number Theory 11R59, 11R58, 11G09 |
| url | https://arxiv.org/abs/2503.00288 |