Article contents
Minimal theta functions
Published online by Cambridge University Press: 18 May 2009
Extract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
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
- Copyright
- Copyright © Glasgow Mathematical Journal Trust 1988
References
1.Cassels, J. W. S., On a problem of Rankin about the Epstein zeta function, Proc. Glasgow Math. Assoc. 4 (1959), 73–80. (Corrigendum, ibid. 6 (1963), 116.)CrossRefGoogle Scholar
2.Delone, B. N. and Ryškov, S. S., A contribution to the theory of the extrema of a multidimensional ζ-function, Dokl. Akad. Nauk SSSR 173 (1963), 523–524. (Translated as Soviet Math. Dokl. 8 (1967), 499–503.)Google Scholar
3.Delone, B. N. and Ryškov, S. S., Extremal problems in the theory of positive definite quadratic forms, Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday, I, Trudy Mat. Inst. Steklov. 112 (1971), 203–223, 387.Google Scholar
4.Diananda, P. H., Notes on two lemmas concerning the Epstein zeta-function, Proc. Glasgow Math. Assoc. 6 (1964), 202–204.CrossRefGoogle Scholar
5.Ennola, Viekko, A lemma about the Epstein zeta function, Proc. Glasgow Math. Assoc. 6 (1964), 198–201.CrossRefGoogle Scholar
6.Ennola, Veikko, On a problem about the Epstein zeta-function, Proc. Cambridge Philos, Soc. 60 (1964), 855–875.CrossRefGoogle Scholar
7.Rankin, R. A., A minimum problem for the Epstein zeta function, Proc. Glasgow Math. Assoc. 1 (1953), 149–158.CrossRefGoogle Scholar
8.Ryškov, S. S., On the question of the final ζ-optimality of lattices that yield the densest packing of n-dimensional balls, Sibirsk. Mat. Ž. 14 (1973), 1065–1075, 1158.Google Scholar
9.Sandakova, N. N., On the theory of ζ-functions of three variables, Dokl. Akad. Nauk SSSR 175 (1967), 535–538.Google Scholar
You have
Access
- 67
- Cited by