Kloosterman sums with primes and the solvability of one congruence with inverse residues — II

Abstract

In the paper, we continue to study the distribution of inverse residues to given modulus. Earlier, the author obtained a series of non-trivial estimates for incomplete Kloosterman sums over prime numbers with an arbitrary modulus $q$. One of the applications of such estimates are some assertions concerning the distribution of inverse residues $\overline{p}$ to prime numbers lying in a "short" segment: $p\overline{p}\equiv 1\pmod{q}$, $1<p\leqslant N$, $N\leqslant q^{1-\delta}$, $\delta>0$, and, more general, concerning the distribution of the quantities $g(p) = a\overline{p}+bp$ with respect to modulus $q$, where $a,b$ are some integers, $(ab,q)=1$.

Another application is connected with the problem of the representation of a given residue $m\pmod{q}$ by the sum $g(p_{1})+\ldots+g(p_{k})$ for fixed $a,b$ and $k\geqslant 3$, in primes $1<p_{1},\ldots,p_{k}\leqslant N$. For the number of such representations, the author have found the formula, where the behavior of the expected main term is controlled by some analogous of the "singular series" that appears in classical circle method, that is, by some function $\kappa$ depending on $q$ and the tuple $k,a,b,m$. For fixed $k,a,b,m$, this function is multiplicative with respect to $q$. In the case when $q$ is not divisible by 2 or 3, this function is strictly positive, and therefore the formula for the number of the representations becomes asymptotic.

In this paper, we study the behavior of $\kappa$ for $q = 3^{n}$. It appears that, for any $n\geqslant 1$, $k\geqslant 3$, there exist the "exceptional" triples $a,b,m$ such that $\kappa = 0$.
The main purpose is to describe all such triples and to obtain the lower estimate for $\kappa$ for all non-exceptional triples.