Concentration points on two and three dimensional modular hyperbolas and applications

Details Concentration points on two and three dimensional modular hyperbolas and applications Concentration points on two and three dimensional modular hyperbolas and applications

2010-07-09

arxiv.org/abs/1007.1526v2

Mathematics

Number Theory

12 pages

Scientific paper

Let $p$ be a large prime number, $K,L,M,\lambda$ be integers with $1\le M\le p$ and ${\color{red}\gcd}(\lambda,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$ xy\equiv\lambda \pmod p, \qquad K+1\le x\le K+M,\quad L+1\le y\le L+M $$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\equiv\lambda\pmod p, \quad L+1\le x,y,z\le L+M. $$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M

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