Kloosterman sums with primes to composite moduli

Abstract

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

Dedicated to Dmitry Aleksandrovich Popov on the occasion with his 80th anniversary. 23 pages

Cited by in corpus (1)

• Kloosterman sums with primes and solvability of one congruence with inverse residues. I