The congruence x1x2≡x3x4(modm) and mean values of character sums

For any positive integer m we obtain the asymptotic formula,|B∩V′|=|B′|ϕ(m)+O(8ν(m)τ(m)(logm)3(loglogm)7|B|), for the number of solutions of the congruence x1x2≡x3x4(modm) with coordinates relatively prime to m   in a box BB of arbitrary size and position. We also obtain an upper bound for a fourth-order character sum moment,1ϕ(m)∑χ≠χ0|∑x=a+1a+Bχ(x)|4≪8ν(m)τ(m)(logm)3(loglogm)7B2.