Abstract:
Suppose that $p$ is a large prime. In this paper, we prove that, for any natural number $N<p$ the following estimate holds:
$$
\max_{(a,p)=1}\biggl|\sum_{q\le N}e^{2\pi iaq^*/p}\biggr|\le(N^{15/16}+N^{2/3}p^{1/4})p^{o(1)},
$$
where $q$ is a prime and $q^*$ is the least natural number satisfying the congruence $qq^*\equiv1\,(\operatorname{mod}p)$. This estimate implies the following statement: if $p>N>p^{16/17+\varepsilon}$, where $\varepsilon>0$, and if we have $\lambda\not\equiv0\,(\operatorname{mod}p)$, then the number $J$ of solutions of the congruence
$$
q_1(q_2+q_3)\equiv\lambda\quad(\operatorname{mod}p)
$$
for the primes $q_1,q_2,q_3\le N$ can be expressed as
$$
J=\frac{\pi(N)^3}p(1+O(p^{-\delta})),\qquad \delta=\delta(\varepsilon)>0.
$$
This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition $p>N>p^{38/39+\varepsilon}$ was required.
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