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Thursday, October 15, 2015

**Abstract:** We estimate the sums $\sum_{c\leq x} \frac{S(m,n,c,\chi)}{c},$ where $S(m,n,c,\chi)$ are Kloosterman sums of half-integral weight on the modular group associated to the multiplier system $\chi$ for the Dedekind eta function. Our estimates are uniform in $m,n,$ and $x$ in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. We approach the problem via an analogue of Kuznetsov's trace formula; among other things this requires us to develop mean value estimates for coefficients of Maass cusp forms of weight $1/2$ and uniform estimates for $K$-Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's formula for the partition function $p(n)$.