The congruence x(1)x(2)x(3)x(4) (mod p), the equation x(1)x(2)=x(3)x(4), and mean values of character sums

We obtain the asymptotic formulae \B boolean AND V\=(\B\/p)+O(root\B\ log(2) p) for the number of solutions of the congruence x(1)x(2)=x(3)x(4) (mod p) in a box B of arbitrary size and position, and N(B)=(12/pi(2)) B-2 log B + CB2 + O(B-19/13 log(7/13) B), with C given explicitly, for the number of solutions of the diophantine equation x(1)x(2)=x(3).v(4) with 1 less than or equal to x(i) less than or equal to B. We also obtain the upper bound for fourth order character sum moments, 1/(p-1)Sigma(x not equal xv)\Sigma(x=a+1)(a+B) X(x)(4) much less than B(2)log(2)p. (C) 1996 Academic Press, Inc.