Two problems related to the non-vanishing of $L (1, \chi )$

Paolo Codecà; Roberto Dvornicich; Umberto Zannier

doi:10.5802/jtnb.218

Two problems related to the non-vanishing of

L (1, χ)

Paolo Codecà ; Roberto Dvornicich ; Umberto Zannier

Journal de Théorie des Nombres de Bordeaux, Tome 10 (1998) no. 1, pp. 49-64.

Résumé
Abstract

Dans cet article, nous étudions deux problèmes à priori assez éloignés, l’un se rapportant à la géométrie diophantienne, et l’autre à l’analyse de Fourier. Tous deux induisent en réalité des questions très proches, relatives à l’étude du rang de matrices dont les coefficients sont nuls ou égaux à $((x y / q)), (0 \leq x, y < q)$ , où $((x)) = x - [x] - 1 / 2$ désigne la partie fractionnaire “centrée” de $x$ . L’étude de ces rangs est liée au problème d’annulation des fonctions $L$ de Dirichlet au point $s = 1$ .

We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or $((x y / q)), (0 \leq x, y < q)$ , where $((u)) = u - [u] - 1 / 2$ denotes the “centered” fractional part of $x$ . These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet $L$ -functions at $s = 1$ .

MR 1827285 | Zbl 0929.11003

DOI : https://doi.org/10.5802/jtnb.218

@article{JTNB_1998__10_1_49_0,
     author = {Codec\`a, Paolo and Dvornicich, Roberto and Zannier, Umberto},
     title = {Two problems related to the non-vanishing of $L (1, \chi )$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {49--64},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {1},
     year = {1998},
     doi = {10.5802/jtnb.218},
     mrnumber = {1827285},
     zbl = {0929.11003},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.218/}
}

Paolo Codecà; Roberto Dvornicich; Umberto Zannier. Two problems related to the non-vanishing of $L (1, \chi )$. Journal de Théorie des Nombres de Bordeaux, Tome 10 (1998) no. 1, pp. 49-64. doi : 10.5802/jtnb.218. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.218/

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