Transformations of Theta-Functions and Analogues of Dedekind Sums
Goldberg, Larry A.
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https://hdl.handle.net/2142/68188
Description
Title
Transformations of Theta-Functions and Analogues of Dedekind Sums
Author(s)
Goldberg, Larry A.
Issue Date
1981
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Certain arithmetical sums arise in the transformation formulae for the logarithms of the classical theta-functions. These sums, which we refer to as Berndt sums, are analogous to Dedekind sums, which arise in a similar manner from the transformation formula for the logarithm of the Dedekind eta-function. In this thesis, we deduce a number of properties of these sums, and demonstrate the central role they play in a variety of results concerning the theta-functions.
First, we derive convergent infinite series for the Fourier coefficients of the reciprocals of the theta-functions. These series representations are similar to Rademacher's series for the partition function p(n), and are simpler than previous representations obtained by Zuckerman. Identities for the coefficients in the series are then deduced. These identities resemble those given by Selberg and Whiteman for the coefficients in Rademacher's series. Multiplicative properties of these coefficients follow from our identities.
Our next application of Berndt sums involves the well-known result that r(,s)(m), the number of representations of m as a sum of s squares, is equal to the singular series (rho)(,s)(m) for 5 (LESSTHEQ) s (LESSTHEQ) 8. We represent the multiplier system of (theta)('s)(z), which is the generating function for r(,s)(m), in terms of Berndt sums instead of Jacobi symbols. This leads to a simpler and more elegant proof than the classical one.
After this, we obtain trigonometric representations for the Berndt sums. This allows us to evaluate certain classes of conditionally convergent double series in terms of these sums. We then derive a number of properties of the Berndt sums, including reciprocity formulae, three-term relations and the Petersson-Knopp identities. Finally, we show how some formulae obtained by Hardy can be represented in terms of these sums, and, in fact, can all be derived from just one transformation formula.
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