New estimate for Kloosterman sums with primes (Q2197236)

Let \(q\) be a positive integer. Let \(a\) be an integer such that \((a,q)=1\). For an integer \(n\) with \((n,q)=1\), let \(\overline{n}\) denote the inverse of \(n\) modulo \(q\), that is \(n\overline{n}\equiv1\bmod q\). For \(X>0\), the Kloosterman sum over primes is given by \begin{align*} T_{q}(X):=\sum_{\substack{n\leq X\\ (n,q)=1}}\Lambda(n)e_q(a\overline{n}), \end{align*} where \(\Lambda(n)\) is the von Mangoldt function, and \(e_q(n):=\exp\left(\frac{2\pi i n}{q}\right)\).\\ In the paper under review, the authors prove nontrivial estimates for \(T_q(X)\), for large values of the modulus \(q\), when \(q^{1/2+\varepsilon}\leq X\leq q^{2/3}\) and when \(q^{5/8+\varepsilon}\leq X\leq q^{3/2}\). The results combined with the estimates proved by \textit{Fouvry} and \textit{I. E. Shparlinski} [Acta Arith. 150, No. 3, 285--314 (2011; Zbl 1243.11093)] for the range \(q^{3/4}\ll X\ll q^{4/3}\) is given in the following main result of the paper. Theorem. Let \(0<\delta<1/100\) and \(0<\varepsilon\leq 1/6\) be arbitrarily small fixed numbers, let \(q\geq q_0(\delta, \varepsilon)\), \(a\) be an integers with \((a,q)=1\), and let \(q^{1/2+\varepsilon}\leq X\leq q^{3/2}\). Then the estimate \begin{align*} T_{q}(X)\ll Xq^{\delta}\Delta \end{align*} holds where \begin{align*} \Delta=\begin{cases} (Xq^{-1/2})^{-\alpha}\quad &\text{if \(q^{1/2+\varepsilon}\leq X\leq q^{\beta}\)},\\ (Xq^{-5/8})^{-4/19}\quad &\text{if \(q^{\beta}\leq X\leq q^{21/26}\)},\\ (Xq^{-1/2})^{-1/8}\quad &\text{if \(q^{21/26}\leq X\leq q^{9/10}\)},\\ (Xq^{-3/4})^{-1/3}\quad &\text{if \(q^{9/10}\leq X\leq q^{12/13}\)},\\ X^{-1/16}\quad &\text{if \(q^{12/13}\leq X\leq q^{3/2}\)}, \end{cases} \end{align*} and \begin{align*} \alpha=\frac{1}{8}(29-5\sqrt{33})=0.034648\dots, \quad\quad \beta=\frac{4077+95\sqrt{33}}{7116}=0.649625\dots\, . \end{align*}