Theta Functions in Complex Analysis and Number Theory

  • Hershel M. FarkasEmail author
Part of the Developments in Mathematics book series (DEVM, volume 17)


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.

Key words

Theta functions triangular numbers partitions Riemann map Picard. 


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  1. 1.
    H.M. Farkas, On an Arithmetical Function, Ramanujan Journal vol. 8 no. 3 pp. 309–315 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    H.M. Farkas, Y. Godin, Logrithmic Derivatives of Theta Functions, Israel Jnl. of Math. vol. 148, pp. 253–265 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    H.M. Farkas Sums of Squares and Triangular Numbers Online Journal of Analytic Combinatorics vol. 1 (2006).Google Scholar
  4. 4.
    H.M. Farkas, I. Kra, Riemann Surfaces, Springer-Verlag (1980).Google Scholar
  5. 5.
    H.M. Farkas, I. Kra, Theta Constants Riemann Surfaces and the Modular Group, AMS Grad Studies in Math, vol. 37 (2001).Google Scholar
  6. 6.
    H.E. Rauch, H.M. Farkas, Theta Functions with Applications to Riemann Surfaces, Williams and Wilkins, Balt. Md. (1974).zbMATHGoogle Scholar
  7. 7.
    C.H. Clemmens, A Scrapbook of Complex Curve Theory, Plenum Press (1980).Google Scholar
  8. 8.
    H.M. Farkas, On an Arithmetical Function II, Contemp. Math. 382, Complex Analysis and Dynamical Systems II (2005).Google Scholar
  9. 9.
    R.D.M. Accola, Theta Functions and Abelian Automorphism Groups, Lecture Notes in Math, Springer-Verlag (1975).Google Scholar
  10. 10.
    G.E. Andrews, Further Problems on Partitions, American Math Monthly, May 1987, pp. 437–439.Google Scholar
  11. 11.
    F.G. Garvan, Shifted and shiftless partition identities, In: Number theory for the millennium, II (Urbana, IL, 2000), pp. 75–92, A.K. Peters, Natick, MA, 2002.Google Scholar

Copyright information

© Springer-Verlag New York 2008

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalem 91904Israel

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