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

Paolo Codecà; Roberto Dvornicich; Umberto Zannier

Journal de théorie des nombres de Bordeaux (1998)

- Volume: 10, Issue: 1, page 49-64
- ISSN: 1246-7405

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topCodecà, Paolo, Dvornicich, Roberto, and Zannier, Umberto. "Two problems related to the non-vanishing of $L (1, \chi )$." Journal de théorie des nombres de Bordeaux 10.1 (1998): 49-64. <http://eudml.org/doc/248160>.

@article{Codecà1998,

abstract = {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 $((xy/q)), (0 \le x, y < q)$, 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 $s = 1$.},

author = {Codecà, Paolo, Dvornicich, Roberto, Zannier, Umberto},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {arithmetic functions; Dirichlet -functions; Fourier series; ranks of matrices},

language = {eng},

number = {1},

pages = {49-64},

publisher = {Université Bordeaux I},

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

url = {http://eudml.org/doc/248160},

volume = {10},

year = {1998},

}

TY - JOUR

AU - Codecà, Paolo

AU - Dvornicich, Roberto

AU - Zannier, Umberto

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

JO - Journal de théorie des nombres de Bordeaux

PY - 1998

PB - Université Bordeaux I

VL - 10

IS - 1

SP - 49

EP - 64

AB - 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 $((xy/q)), (0 \le x, y < q)$, 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 $s = 1$.

LA - eng

KW - arithmetic functions; Dirichlet -functions; Fourier series; ranks of matrices

UR - http://eudml.org/doc/248160

ER -

## References

top- [1] W.A. Adkins & S.H. Weintraub - Algebra, An Approach via Module Theory, Springer Verlag (1992). Zbl0768.00003MR1181420
- [2] P. Cellini - A general commutative descent algebra, Journal of Algebra175 (1995), 990-1014. Zbl0832.20060MR1341755
- [3] H. Davenport - On some infinite series involving arithmetical functions, QuarterlyJournal of Mathematics8 (1937), 8-13. Zbl0016.20105JFM63.0906.01
- [4] S.L. Segal - On an identity between infinte series and arithmetic functions, Acta ArithmeticaXXVIII (1976), 345-348. Zbl0319.10050MR387222
- [5] L.C. Washington - Introduction to Cyclotomic Fields, Springer-Verlag (1982). Zbl0484.12001MR718674

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