Character Sums and Congruences with n!

Abstract

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer $n\ll p^{1/2+\epsilon },suchthatn!isaprimitiverootmodulop.Wealsoshowthateverynonzerocongruenceclassa\not\equiv 0(\mathrm{mod}p)canberepresentedasaproductof7factorials,a\equiv n_1!...n_7!(\mathrm{mod}p),where$\max \{n_i | i=1,... 7\}=O(p^{11/12+ε}), and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!,with\max \{n_1,n_2,n_3,n_4\}=O(p^{6/7+\epsilon })$ represent ``almost all''residue classes modulo p, and that products of 3 factorials n_1!n_2!n_3! with \max\{n_1, n_2, n_3\}=O(p^{5/6+ε})$ are uniformly distributed modulo p.

20 pages. Trans. Amer. Math. Soc. (to appear)