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Minimal theta functions

Published online by Cambridge University Press:  18 May 2009

Hugh L. Montgomery
Hugh L. Montgomery
Affiliation:
University of Michigan, Ann Arbor, Michigan 48109, U.S.A.
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Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.

Among such forms, let . The Epstein zeta function of f is denned to be

Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,

We prove a corresponding result for theta functions. For real α > 0, let

This function satisfies the functional equation

(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 1 , January 1988 , pp. 75 - 85
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

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