The Congruencex1x2≡x3x4(modp), the Equationx1x2≡x3x4, and Mean Values of Character Sums

We obtain the asymptotic formulae ∣B∩V∣=(∣B∣/p)+O(√∣B∣log2p) for the number of solutions of the congruencex1x2≡x3x4(modp) in a box B of arbitrary size and position, andN(B)=(12/π2)B2logB+CB2+O(B19/13log7/13B), withCgiven explicitly, for the number of solutions of the diophantine equationx1x2≡x3x4with 1≤xi≤B. We also obtain the upper bound for fourth order character sum moments, 1/(p−1) ∑χ≠χo∣ ∑a+Bx=a+1χ(x)∣4⪡B2log2p.