On the Sato-Tate conjecture for Kloosterman sums
Let \(S(a, b; q)\) be the usual Kloosterman sum, and let \(S(1, m; p)= 2\sqrt {p} \cos \vartheta_{p,m}\) with \(0\leq \vartheta_{p,m}\leq \pi\). The Sato-Tate conjecture is that, for fixed \(m\neq 0\), the angles \(\vartheta_{p,m}\) have a distribution function \({2\over \pi} \sin^2 \vartheta\) as the prime \(p\) varies. Here it is shown that, for each prime \(p\in [x,2x ]\) the angles \(\vartheta_{p,q}\) \((0<q< p)\) and \(\vartheta_{p, mq^{-2}}\) \((q\in [x,2x ])\) follow the Sato-Tate distribution. As corollaries one finds that a positive proportion of pairs \(p,q\leq x\) of primes have \(|S(1, m; pq)|\geq (0.16) 4\sqrt {pq}\), and for fixed \(\varepsilon>0\), a positive proportion have \(|S(1, m; pq)|\leq \varepsilon \sqrt {pq}\). The constant \(0.16\) arises because the Sato-Tate distribution accords a probability greater than \({1\over 2}\) to the event \(\cos \vartheta\geq 0.4\). The proofs use estimates for multiple exponential sums, for which a certain amount of algebraic geometry is required. They also depend on a sieve identity (of Buchstab type) to handle the summation over primes.
scientific article; zbMATH DE number 784019