Theta functions in complex analysis and number theory

Hershel M. Farkas

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

10 Scopus citations

Abstract

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.

Original languageEnglish
Title of host publicationSURVEYS IN NUMBER THEORY
EditorsKRISHNASWAMI ALLADI
Pages57-87
Number of pages31
DOIs
StatePublished - 2008

Publication series

NameDevelopments in Mathematics
Volume17
ISSN (Print)1389-2177

Keywords

  • Partitions
  • Picard
  • Riemann map
  • Theta functions
  • Triangular numbers

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