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

2019/11/28 by Changa, M. E., Korolev, M. A.
#11A07 #11L05 #11L07 #11L20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.12589

Abstract

In the paper, we establish a new estimate for Kloosterman sum over primes with respect to an arbitrary modulus q. This estimate together with some recent results of the second author are applied to the problem of solvability of the congruence g(p₁) + … + g(p_k) ≡ m\pmodq in prime variables p₁,…, p_k≤ N, N≤ q^1-δ. Here g(x) ≡ ax+bx\pmodq, where a,b,m and k≥ 3 are arbitrary integers, (ab,q)=1. The main result of the paper gives an asymptotic formula for the number of solutions for the case when q is coprime to 6. In this version, we correct some typos and small errors.

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