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Minimal theta functions

Published online by Cambridge University Press:  18 May 2009

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
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

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