Uniform bounds for Kloosterman sums of half-integral weight with applications

Abstract

Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to $x$ with implied constants depending on $m$ and $n$. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in $x$, $m$ and $n$. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of $x$ with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in $x$, $m$ and $n$. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to $m$ and $n$, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function $p(n)$. We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo $3$, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.