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Bulletin of the London Mathematical Society 2005 37(1):45-53; doi:10.1112/S0024609304003716
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© London Mathematical Society

Gaps Between the Zeros of Epstein's Zeta-Functions on the Critical Line

M. Jutila and K. Srinivas

Department of Mathematics, University of Turku FIN-20014 Turku, Finland; jutila{at}utu.fi
Institute of Mathematical Sciences Tharamani P. O., Chennai-600113, India; srini{at}imsc.res.in

Received 3 October 2003. Revision received 17 February 2004.

It is proved that Epstein's zeta-function {zeta}Q(s), related to a positive definite integral binary quadratic form, has a zero 1/2+i{gamma} with T ≤ {gamma} ≤ T + T5/11+{varepsilon} for sufficiently large positive numbers T. This improves a classical result of H. S. A. Potter and E. C. Titchmarsh (Proc. London Math. Soc. (2) 39 (1935) 372–384). 2000 Mathematics Subject Classification 11E45 (primary), 11M41 (secondary).


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