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arXiv · 2212.00366

On linear independence of Dirichlet $L$ values

Sanoli Gun·Neelam Kandhil·Patrice Philippon
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Abstract

The study of linear independence of $L(k, \chi)$ for a fixed integer $k>1$ and varying $\chi$ depends critically on the parity of $k$ vis-\`a-vis $\chi$. This has been investigated by a number of authors for Dirichlet characters $\chi$ of a fixed modulus and having the same parity as $k$.The focal point of this article is to extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus (consequently a single number field lurking in the background). This entails a very careful and hands-on dealing with the arithmetic of compositum of number fields which we undertake in this work. Our results extend earlier works of the first author with Murty-Rath as well as works of Okada, Murty-Saradha and Hamahata.

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Sanoli Gun, Neelam Kandhil, Patrice Philippon. 2022-12-01. On linear independence of Dirichlet $L$ values. https://doi.org/10.1016/j.jnt.2022.09.016

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