We estimate the sums c≤x S(m, n, c, χ) c ,where the S(m, n, c, χ) are Kloosterman sums of half-integral weight on the modular group. 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. 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).

Abstract. Lehner's 1949 results on the j-invariant showed high divisibility of the function's coefficients by the primes p ∈ {2, 3, 5, 7}. Expanding his results, we examine a canonical basis for the space of level p modular functions holomorphic at the cusp 0. We show that the Fourier coefficients of these functions are often highly divisible by these same primes.

No abstract

Building on work of Hardy and Ramanujan, Rademacher proved a well-known formula for the values of the ordinary partition function $p(n)$. More recently, Bruinier and Ono obtained an algebraic formula for these values. Here we study the smallest parts function introduced by Andrews; $\operatorname{spt}(n)$ counts the number of smallest parts in the partitions of $n$. The generating function for $\operatorname{spt}(n)$ forms a component of a natural mock modular form of weight $3/2$ whose shadow is the Dedekind eta function. Using automorphic methods (in particular the theta lift of Bruinier and Funke), we obtain an exact formula and an algebraic formula for its values. In contrast with the case of $p(n)$, the convergence of our expression is non-trivial, and requires power savings estimates for weighted sums of Kloosterman sums for a multiplier in weight $1/2$. These are proved with spectral methods (following an argument of Goldfeld and Sarnak).Comment: 21 pages, 2 figures. Version 3 corrects an error in normalization which affects the constants in Theorem 3, Theorem 8, and Lemma 9. These changes will appear as a corrigendum to the published versio

We prove an exact formula for the second moment of Rankin-Selberg L-functions L( 1 2 , f × g) twisted by λ f (p), where g is a fixed holomorphic cusp form and f is summed over automorphic forms of a given level q. The formula is a reciprocity relation that exchanges the twist parameter p and the level q. The method involves the Bruggeman/Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.