Sums of moduli of exponential sums (Q1880042)

Let \(Kl(a,b; n)\) be the usual Kloosterman sum, and let \[ A^*(x)= \sum_{n\leq x} {|Kl(1,1;n)|\over 2^{\omega(n)}\sqrt{n}},\;\widetilde A(x)= \sum_{n\leq x} {|Kl(1,1;n)|\over \sqrt{n}}. \] It is shown that there is a constant \(\delta> 5/12\) such that \[ {x\over\log x} \exp((\log\log x)^\delta)\ll A^*(x)\ll x\Biggl({\log\log x\over\log x}\Biggr)^{1-4/3\pi} \] and similarly \[ {x\over\log x}\exp((\log\log x)^\delta)\ll\widetilde A(x)\ll x\Biggl({\log\log x\over\log x}\Biggr)^{1-8/3\pi}. \] Such estimates were first considered by \textit{C. Hooley} [Mathematika, Lond. 11, 39--49 (1964; Zbl 0123.25802)] forty years ago, who gave, in effect, a bound \[ \widetilde A(x)\ll x(\log x)^{\sqrt{2}-1}(\log\log x)^c. \] Analogous results for certain more general exponential sums in one variable are also presented. The proof is a technical masterpiece, combining a result of the second author [Duke Math. J. 95, No. 2, 227--240 (1998; Zbl 0958.11056)] on the distribution of exponential sums, with a large sieve inequality, and an analysis of the contributions from ``small'' and ``large'' prime factors.