Kloosterman Sums with Primes and Solvability of a Congruence with Inverse Residues

The problem of the solvability of the congruence $g(p_1)+\dots +g(p_k)\equiv m\pmod {q}$ in primes $p_1,\dots ,p_k\leq N$, $N\leq q^{1-\gamma }$, $\gamma >0$, is addressed. Here $g(x)\equiv a\overline {x}+bx\pmod {q}$, $\overline {x}$ is the inverse of the residue $x$, i.e., $\overline {x}x\equiv 1\pmod {q}$, $q\geq 3$, and $a$, $b$, $m$, and $k\geq 3$ are arbitrary integers with $(ab,q)=1$. The analysis of this congruence is based on new estimates of the Kloosterman sums with primes. The main result of the study is an asymptotic formula for the number of solutions in the case when the modulus $q$ is divisible by neither $2$ nor $3$.