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Lithuanian Mathematical Journal

, Volume 58, Issue 2, pp 198–211 | Cite as

On the uniform distribution of zero ordinates of Epstein zeta-functions

  • Christof Schmeller
Article
  • 33 Downloads

Abstract

Let \( \zeta \left(s;\mathcal{Q}\right) \) be the Epstein zeta-function associated with an integral quadratic form \( \mathcal{Q}\left[y\right]={\mathrm{y}}^{\mathrm{t}}\mathcal{Q}\mathrm{y} \) for y ϵ ℤ n with nontrivial zeros ρ = β + iy. We prove that \( {\Sigma}_{\left[\upgamma \right]<{T}^{X^{\rho }}}={c}_xT+O\left(\log T\right) \) for all Epstein zeta functions analogously to Landau’s explicit formula in the case of the Riemann zeta function. It follows that under the assumption \( {\Sigma}_{\left[\upgamma \right]<T}\left|\beta -n/4\right|=o\left(N\left(T;\mathcal{Q}\right)\right), \) where \( N\left(T;\mathcal{Q}\right) \) counts the nontrivial zeros for |γ| < T, the ordinates of Epstein zeta zeros are uniformly distributed modulo one.

Keywords

Epstein zeta function nontrivial zeros uniform distribution 

MSC

11M41 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Christof Schmeller
    • 1
  1. 1.ReinheimGermany

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