A lower bound for classical Kloosterman sums and an application

2024/06/18 by Baier, Stephan, Das, Jishu, Mahajan, Jewel
#11F72 #FOS: Mathematics #Number Theory (math.NT) #Primary 11L05

paper · doi:10.48550/arxiv.2406.13013

Abstract

We present a lower bound for the classical Kloosterman sum S(a,b;c) where (ab,c)=1 and c is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight k and level N are permitted to vary independently. Using this modified version, we get a lower bound concerning the trace of the Hecke operator T_n acting on the space S(N, k), of cusp forms of weight k and level N for (n, N) =1.

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