ON ZEROS AND c-VALUES OF EPSTEIN ZETA-FUNCTIONS

Author:

Takashi NAKAMURA$, {\L}ukasz PA\'NKOWSKI

2010 Mathematics Subject Classification:

10C15, 11M41.

Keywords:

hybrid universality, zeros and c-values of Epstein zeta-functions.

Abstract:

Let {\mathcal{Q}} be a positive definite n\times n matrix and \zeta (s; {\mathcal{Q}}) be the Epstein zeta-function associated with {\mathcal{Q}}. In the present paper, we prove that, for arbitrary given complex number c, the equation \zeta (s; {\mathcal{Q}})=c has at least CT, for some positive constant~C, solutions in the region \Re s > \frac{n-1}{2} when n\geq 4 is even and {\mathcal{Q}} satisfies certain conditions. As a corollary,  we show that \zeta (s; I_{2k}), where {\mathbb{N}} \ni k \ne 1,2,4 and I_n is the n-dimensional unit matrix, have complex zeros in the strip k-\frac{1}{2} < \Re s < k.

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Naka-Pank-2013

Vol. 8 (16), 2013