On a generalisation of a Lehmer problem

Abstract

We consider a generalisation of the classical Lehmer problem about the distribution of modular inverses in arithmetic progression, introduced by E. Alkan, F. Stan and A. Zaharescu. Using bounds for multiplicative character sums instead of bounds for Kloosterman sums traditionally applied to this kind of problem, we improve their results in several directions.

On congruences with products of variables from short intervals and applications

“…For large values of h, one can use bounds of Kloosterman sums (for ν = 2, 3) and multiplicative character sums (for ν ≥ 4) to obtain various asymptotic formulas for J ν (p, h, s; λ), see [14,15,21,24,25]. However, this approach does not give any nontrivial estimates for small values of h, and thus Chan and Shparlinski [5], for ν = 2, have employed methods of additive combinatorics, namely some results of Bourgain [3], in order to obtain a nontrivial upper bound on J ν (p, h, s; λ) for any h.…”

On Solutions to Some Polynomial Congruences in Small Boxes

“…As we have mentioned the case of just one congruence (2) has been considered in [9,10], so we always assume that s ≥ 1 (and thus n ≥ 3). Let k = min{k i,j : i = 1, .…”

Modular hyperbolas

“…For the dimension s = 3 it is based on Theorem 16. For s 4 the proof is based on various bounds for multiplicative character sums in the same style as in [169,176].…”