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Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime
Authors:M. Z. Garaev
Affiliation:(1) Universidad Nacional Autónoma de México, Morelia, México
Abstract: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,pX, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, yp?(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, yp?(log p)1+o(1) and y* is defined by yy* ≡ 1 (mod p).
Keywords:2000 Mathematics Subject Classifications: 11A07   11L40
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