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

We obtain the asymptotic formulae midimage∩Vmid=(midimagemid/p)+O(√imagelog2p) for the number of solutions of the congruencex1x2≡x3x4(modp) in a box image 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≠χomid ∑a+Bx=a+1χ(x)mid4much less-thanB2log2p.