Minimal theta functions

@article{Montgomery1988MinimalTF,
  title={Minimal theta functions},
  author={Hugh L. Montgomery},
  journal={Glasgow Mathematical Journal},
  year={1988},
  volume={30},
  pages={75 - 85}
}
  • H. Montgomery
  • Published 1 January 1988
  • Mathematics
  • Glasgow Mathematical Journal
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).) 
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References

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    Proceedings of the Glasgow Mathematical Association
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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]
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    Proceedings of the Glasgow Mathematical Association
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Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shape with y > 0. We write (for s > 1) The function Zn(s) may be analytically
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    Proceedings of the Glasgow Mathematical Association
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In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of
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    Proceedings of the Glasgow Mathematical Association
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1. We use Cassels's notation and define h (m, n), Q (m, n), Zh (s), Zh (1) – ZQ (1) and G (x, y) as in [1]. Rankin [5] proved that the Epstein zeta-function Zh (s) satisfies, for s ≧ 1·035, the
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