Kloosterman sums with primes to composite moduli

Abstract: We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type [ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi i}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad p\overline{p}\equiv 1\pmod{q}, ] which is non-trivial in the case when $q^{\,3/4+\varepsilon}\leqslant X\ll q^{\,3/2}$. We also apply this estimate to the proof of solvability of some congruences with inverse prime residues $\pmod{q}$.