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A lemma about the Epstein zeta-function

Published online by Cambridge University Press:  18 May 2009

Veikko Ennola
Veikko Ennola
Affiliation:
University of TurkuFinland
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Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shape

with y > 0. We write (for s > 1)

The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 6 , Issue 4 , July 1964 , pp. 198 - 201
Copyright
Copyright © Glasgow Mathematical Journal Trust 1964

References

REFERENCES

1.Cassels, J. W. S., On a problem of Rankin about the Epstein Zeta-function, Proc. Glasgow Math. Assoc. 4 (1959), 73–80.CrossRefGoogle Scholar
2.Epstein, P., Zur Theorie allgemeiner Zetafunktionen, Math. Ann. 56 (1903), 615–644.CrossRefGoogle Scholar
3.Rankin, R. A., A minimum problem for the Epstein Zeta-function, Proc. Glasgow Math. Assoc. 1 (1953), 149–158.CrossRefGoogle Scholar