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On a problem of Rankin about the Epstein zeta-function

Published online by Cambridge University Press:  18 May 2009

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Trinity College, Cambridge
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Let

be a positive definite quadratic form with determinant αβ−X2 = 1. A special form of this kind is

We consider the Epstein zeta-function

the series converging for s > 1. For s ≥ 1·035 Rankin [1] proved the following

STatement R.

The sign of equality is needed only when h is equivalent to Q.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 2 , July 1959 , pp. 73 - 80
Copyright
Copyright © Glasgow Mathematical Journal Trust 1959

References

1.Rankin, R. A., A minimum problem for the Epstein zeta-function, Proc. Glasgow Math. Assoc. 1 (1953), 149–158.CrossRefGoogle Scholar
2.Weber, H., Lehrbuch der Algebra III, especially page 526 (Braunschweig, 2te Auflage, 1908).Google Scholar
3.Deuring, M., Zeta-funktionen quadratischer Formen, J. fur reine u, angew. Math. 172 (1935), 226–252.CrossRefGoogle Scholar
4.Watson, G. N., A treatise on the theory of Bessel functions (Cambridge, 1922 (2nd ed. 1944)).Google Scholar
5.Jahnke, E. and Emde, F., Funktionentafeln (Teubner, Leipzig (3rd ed. 1938)).Google Scholar
6.Kronecker, L., Über die Auflösung der Pell'schen Gleichung mittels elliptischer Functionen, Monatsber. d. Kön. Preuss. Akad. d. Wiss. zu Berlin 1863, 44–50 (= Werke IV, 219–226).Google Scholar