A uniform lower bound for classical Kloosterman sums and applications

Abstract: We present an elementary uniform 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 the 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 conditional lower bound for the trace of the operator $(T_n)^t$ for integer $t>1$, where $T_n$ denotes the normalised Hecke operator for natural number $n$ with $(n,N)=1.$