Analytic Number Theory and Multiple Basic Hypergeometric Series

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

The investigator will continue his research on analytic number theory, algebraic combinatorics, and the theory and application of multiple basic hypergeometric series. The investigator recently derived many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n^2 or 4n(n+1) squares, respectively, without using cusp forms. He similarly generalized to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, and Schur functions. The investigator first plans to finish his derivation of the infinite families of explicit exact sums of squares identities (excluding those above) for n^2 or n(n+1) squares, respectively, that he has already shown to exist. These will include explicit identities for an odd number of squares. Next, he plans to prove additional Kac--Wakimoto conjectures for triangular numbers. In the area of algebraic combinatorics, the investigator will develop the combinatorial and number theoretic implications of his new formulas for Ramanujan's tau function that are directly related to the Leech lattice. The above project utilizes explicit combinatorial and analytic methods to study the problem of representing an integer as a sum of squares of integers. This problem is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic ``Fundamenta Nova'' of 1829. He found elegant simple formulas for counting the number of ways that any fixed positive integer n can be written as a sum of 2, 4, 6, or 8 squares. Applications of modular forms to sums of squares really started in 1917 with Mordell. This approach provides abstract formulas for any given number of squares as the Fourier coefficients of modular forms. In 1965 Rankin proved that any one of these formulas, for more than 8 squares, would be highly non-trivial to compute explicitly. Since 1829, explicit exact non-trivial formulas have only been found for up to 32 squares, with an even number of squares much easier than an odd number. The investigator used his combinatorial/elliptic function methods to derive infinite families of expansions of powers of classical theta functions, and the corresponding non-trivial explicit sums of squares formulas. (See http://xxx.lanl.gov/abs/math.NT/0008068). This is the first time that infinite families of non-trivial exact explicit formulas for sums of squares have been found. All of this work gives a new elegant extension of the classical formulas of Jacobi.