2017/08/09 by Dunn, Alexander
#11L05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1708.03003
For m,n>0 and mn<0 we estimate the sums ∑c ≤ x (S(m,n,c,χ))/(c), where the S(m,n,c,χ) are Kloosterman sums attached to a multiplier χ of weight 1/2 on the full modular group. Our estimates are uniform in m, n and x in analogy with the bounds for the case mn<0 due to Ahlgren-Andersen, and those of Sarnak-Tsimerman for the trivial multiplier when m,n>0. In the case mn<0, our estimates are stronger in the mn-aspect than those of Ahlgren-Andersen. We also obtain a refinement whose quality depends on the factorization of 24m-23 and 24n-23 as well as the best known exponent for the Ramanujan-Petersson conjecture.