Number Theory
On the nontrivial zeros of modified Epstein zeta functions
Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 79-81.

We study the zeros of modified Epstein zeta functions having functional equations. The result is that for any δ>0, all but finitely many nontrivial zeros of such a function in {sC:|s12|<δ} are simple and on the critical line. As an immediate consequence of this theorem, all but finitely many nontrivial zeros of many modified Epstein zeta functions are simple and on the critical line.

Nous étudions les zéros des fonctions Zeta modifiées d'Epstein ayant des équations fonctionnelles. Notre résultat est, que pour tout δ>0, tous les zéros non-triviaux d'une telle fonction dans {sC:|s12|<δ} sauf au plus un nombre fini d'entre eux, sont simples et sur la droite critique. Une conséquence immédiate de ce théorème est que tous les zéros non-triviaux, sauf au plus un nombre fini d'entre eux, de beaucoup de fonctions Zeta modifiées d'Epstein sont simples et sur la droite critique.

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DOI: 10.1016/j.crma.2005.11.015
Ki, Haseo 1

1 Department of Mathematics, Yonsei University, Seoul 120–749, South Korea
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Ki, Haseo. On the nontrivial zeros of modified Epstein zeta functions. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 79-81. doi : 10.1016/j.crma.2005.11.015. http://www.numdam.org/articles/10.1016/j.crma.2005.11.015/

[1] Chowla, S.; Selberg, A. On Epstein's zeta-function, J. Reine Angew. Math., Volume 227 (1967), pp. 86-110

[2] Ki, H. All but finitely many nontrivial zeros of the approximations of the Epstein zeta function are simple and on the critical line, Proc. London Math. Soc., Volume 90 (2005), pp. 321-344

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This work was supported by grant No. R01-2005-000-10339-0 from the Basic Research Program of the Korea Science & Engineering Foundation.