A note on signs of Kloosterman sums (Q2880693)

Let \(Kl(a,b;n)\) be the Kloosterman sums and \(\omega(n)\) be the number of distinct prime divisors of \(n\). In this paper ,the author proves the following results.NEWLINENEWLINE Theorem 1.1. There exist \(X_{0}\geq 1,c_{0}>0\) such that, for \(X\geq X_{0}\), we have NEWLINE\[NEWLINE|\{X\leq n<2 X\mid Kl(1,1;n)>0,\mu^{2}=1,\omega(n)\leq 15\}|\geq c_{0}\frac{X}{\log X}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE|\{X\leq n<2 X\mid Kl(1,1;n)<0,\mu^{2}=1,\omega(n)\leq 15\}|\geq c_{0}\frac{X}{\log X}.NEWLINE\]NEWLINE The proofs use an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.