The Linnik conjecture. I. (Q1854794)

Linnik's famous conjecture on sums of Kloosterman sums is the hypothetical inequality \[ \sum_{c\leq x}S(m,n;c)/c \ll x^{\varepsilon }, \] for fixed positive integers \(m\) and \(n\). The point of this conjecture is that significant cancellation is expected in the \(c\)-sum, and remarkable progress in this direction was made by \textit{N. V. Kuznetsov} [Mat. Sb., Nov. Ser. 111, 334--383 (1980; Zbl 0427.10016)] who proved the above inequality with \(\varepsilon =1/6\); another proof of this result was given by \textit{D. Goldfeld} and \textit{P. Sarnak} [Invent. Math. 71, 243--250 (1983; Zbl 0507.10029)] using the Kloosterman sum zeta-function. The author applies the same zeta-function as well to establish Linnik's conjecture in mean for sums involving \(S(1,m;c)\); it is known that a general Kloosterman sum can be written in terms of such sums. More precisely, the mean square of Linnik's sum over \(x\in [N/2,N]\) is of the expected order uniformly for \(m \ll N^2\), and likewise for the mean square over \(m \leq M\) if \(x \ll M \ll x^2\). Also, an analogous result is obtained for a double average in both aspects at the same time.