Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-25T17:34:14.214Z Has data issue: false hasContentIssue false
  • English
  • Français

On the zeros of Epstein's zeta function

Published online by Cambridge University Press:  26 February 2010

H. M. Stark
H. M. Stark
Affiliation:
The Department of Mathematics, The University of Michigan, Ann Arbor, Michigan, U.S.A.
Get access

Extract

Let Q(x, y) = ax2 + bxy + cy2 be a positive definite quadratic form with discriminant d = b2 – 4ac. The Epstein zeta function associated with Q is given by

where Σ′ means the sum is over all pairs (x, y) of integers not both zero, and as usual, s = σ + it.

Type
Research Article
Information
Mathematika , Volume 14 , Issue 1 , June 1967 , pp. 47 - 55
Copyright
Copyright © University College London 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bateman, P. T. and Grosswald, E., “On Epstein's zeta function”, Ada Arithmetica, 9 (1964), 365–373.CrossRefGoogle Scholar
2.Potter, H. S. A. and Titchmarsh, E. C., “The zeros of Epstein's zeta-functions”, Proc. London Math. Soc. (2), 39 (1935), 372–384.CrossRefGoogle Scholar
3.Davenport, H. and Heilbronn, H., “On the zeros of certain Dirichlet series I, II”, Journal London Math. Soc., 11 (1936), 181–185 and 307-312.CrossRefGoogle Scholar
4.Titchmarsh, E. C., The theory of the Riemann zeta-function (Oxford, 1951).Google Scholar
5.Stark, H. M., “On complex quadratic fields with class number equal to one”, Trans. American Math. Soc., 122 (1966), 112–119.CrossRefGoogle Scholar
6.Walfisz, A., Weylsche Exponentialsummen in der neueren Zahlentheorie (Berlin, 1963).Google Scholar