Maass forms and Ramanujan’s third order mock theta function
In 1964, George Andrews proved an asymptotic formula (finite sum of terms) involving generalized Kloosterman sums and the \(I\)-Bessel function for the coefficients of Ramanujan's famous third order mock theta function. Andrews conjectured that these series converge when extended to infinity, and that it they do not converge absolutely. Bringmann and Ono proved the first of these conjectures in 2006. Here we obtain a power savings bound for the error in Andrews' formula, and we also prove the second of these conjectures.
Our methods depend on the spectral theory of Maass forms of half-integral weight, and in particular on a new estimate which we derive for the Fourier coefficients of such forms which gives a power savings in the spectral parameter as compared to results of Duke and Baruch-Mao.
This is a joint work with Scott Ahlgren.
Alexander Dunn, University of Illinois at Urbana-Champaign