Geodesic
Minimal theta functions
Glasgow mathematical journal, Tome 30 (1988) no. 1, pp. 75-85

Voir la notice de l'article provenant de la source Cambridge University Press

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 beRankin [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, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).) 
Montgomery, Hugh L. Minimal theta functions. Glasgow mathematical journal, Tome 30 (1988) no. 1, pp. 75-85. doi: 10.1017/S0017089500007047
@article{10_1017_S0017089500007047,
     author = {Montgomery, Hugh L.},
     title = {Minimal theta functions},
     journal = {Glasgow mathematical journal},
     pages = {75--85},
     year = {1988},
     volume = {30},
     number = {1},
     doi = {10.1017/S0017089500007047},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007047/}
}
TY  - JOUR
AU  - Montgomery, Hugh L.
TI  - Minimal theta functions
JO  - Glasgow mathematical journal
PY  - 1988
SP  - 75
EP  - 85
VL  - 30
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007047/
DO  - 10.1017/S0017089500007047
ID  - 10_1017_S0017089500007047
ER  - 
%0 Journal Article
%A Montgomery, Hugh L.
%T Minimal theta functions
%J Glasgow mathematical journal
%D 1988
%P 75-85
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007047/
%R 10.1017/S0017089500007047
%F 10_1017_S0017089500007047

[1] 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.) Google Scholar | DOI

[2] 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] 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] 4.Diananda, P. H., Notes on two lemmas concerning the Epstein zeta-function, Proc. Glasgow Math. Assoc. 6 (1964), 202–204. Google Scholar | DOI

[5] 5.Ennola, Viekko, A lemma about the Epstein zeta function, Proc. Glasgow Math. Assoc. 6 (1964), 198–201. Google Scholar | DOI

[6] 6.Ennola, Veikko, On a problem about the Epstein zeta-function, Proc. Cambridge Philos, Soc. 60 (1964), 855–875. Google Scholar | DOI

[7] 7.Rankin, R. A., A minimum problem for the Epstein zeta function, Proc. Glasgow Math. Assoc. 1 (1953), 149–158. Google Scholar | DOI

[8] 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] 9.Sandakova, N. N., On the theory of ζ-functions of three variables, Dokl. Akad. Nauk SSSR 175 (1967), 535–538. Google Scholar

Cité par Sources :