1998/01/31 by Barry Brent
Mathematics · #math.NT #msc:11F11
published as Exp. Math., 7 (1998) 257-274 · To appear in "Experimental Mathematics". 25 pages. Section 5 cuts omit "weakly level 1" results, since weakly level 1 => level 1 (as pointed out to me by Rainer Schulze-Pillot.)
arxiv created 1998/03/01 · arxiv updated 2009/11/30
We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for Γ0(2). Numerical evidence indicates that a sharper bound holds for the weights h ≡ 2 . We derive upper bounds for the minimum positive integer represented by level two even positive-definite quadratic forms. Our data suggest that, for certain meromorphic modular forms and p=2,3, the p-order of the constant term is related to the base-p expansion of the order of the pole at infinity, and they suggest a connection between divisibility properties of the Ramanujan tau function and those of the Fourier coefficients of 1/j.