, Volume 331, Issue 1, pp 219-239
Date: 08 Jun 2004

Coefficients of half-integral weight modular forms modulo ℓ j

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Abstract.

Suppose that ℓ≥5 is prime, that j≥0 is an integer, and that F(z) is a half-integral weight modular form with integral Fourier coefficients. We give some general conditions under which the coefficients of F are “well-distributed” modulo ℓ j . As a consequence, we settle many cases of a classical conjecture of Newman by proving, for each prime power ℓ j with ℓ≥5, that the ordinary partition function p(n) takes each value modulo ℓ j infinitely often.

Mathematics Subject Classification (2000): 11F33, 11P83
The first author thanks the National Science Foundation for its support through grant DMS 01-34577. The second author thanks the National Science Foundation for its support through a VIGRE postdoctoral fellowship.
Acknowledgement The authors thank the referee for suggestions which improved the exposition in this paper.