Incomplete exponential sums and incomplete residue systems for congruences

Exponential Sums and Congruences with Factorials

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials n!m! and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials n!m! with max{n,m} < p1/2+ε are uniformly distributed modulo ...

Exponential sums and polynomial congruences along p-adic submanifolds

In this article, we consider the estimation of exponential sums along the points of the reduction mod pm of a p-adic analytic submanifold of Zp . More precisely, we extend Igusa’s stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod pm of a p-adic analytic submanifold of Zp . In addition,...

Congruences and Exponential Sums with the Euler Function

where gcd(a, p) = 1, and N is sufficiently large. Our bounds are nontrivial for a wide range of values of p, starting with p ≥ logN . We remark that although it might be possible to improve on this power of logN , for very small values of p relative to N , it is simply not possible to obtain nontrivial bounds. In fact, it has been shown in Theorem 3.5 of [5] that for any prime number p of size ...

Incomplete Exponential Sums over Finite Fields and Their Applications to New Inversive Pseudorandom Number Generators

An Estimate of Incomplete Mixed Character Sums