Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

摘要

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p ½(log p)1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy * (mod p), where 1 ≤ x, y ≤ p ½(log p)1+o(1) and y * is defined by yy * ≡ 1 (mod p).