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Theta functions in complex analysis and number theory. (English) Zbl 1206.11055
Alladi, Krishnaswami (ed.), Surveys in number theory. New York, NY: Springer (ISBN 978-0-387-78509-7/hbk). Developments in Mathematics 17, 57-87 (2008).
Summary: In these notes we try to demonstrate the utility of the theory of theta functions in combinatorial number theory and complex analysis. The main idea is to use identities among theta functions to deduce either useful number-theoretic information related to representations as sums of squares and triangular numbers, statements concerning congruences, or statements concerning partitions of sets of integers. In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk.
MSC:
11F27Theta series; Weil representation; theta correspondences
11B65Binomial coefficients, etc.
30C20Conformal mappings of special domains
33E05Elliptic functions and integrals