Distribution of modular inverses and multiples of small integers and the Sato--Tate conjecture on average

2006/08/24 by Shparlinski, I. E.
#11A07 #11K38 #11L05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0608596

Abstract

We show that, for sufficiently large integers m and X, for almost all a =1, ..., m the ratios a/x and the products ax, where |x|≤ X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums K_r,s(p) = ∑_x=1^p-1 exp(2 πi (rn + sn^-1)/p), for primes p≤ T is in accordance with the Sato--Tate conjecture.

Related

• On the distribution of Kloosterman sums
• Generalized Sato-Tate and quadratic residues
• On the distribution of modular inverses from short intervals
• Triple sums of Kloosterman sums and the discrepancy of modular inverses
• Kloosterman sums in residue rings