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

For any positive integer m we obtain the asymptotic formula, vertical bar B boolean AND V'vertical bar = vertical bar B'vertical bar/phi(m) + O(8(nu(m))tau(m)(log m)(3)(log log m)(7) root vertical bar B vertical bar), for the number of solutions of the congruence x(1)x(2) equivalent to x(3)x(4) (mod m) with coordinates relatively prime to m in a box B of arbitrary size and position. We also obtain an upper bound for a fourth-order character sum moment, 1/phi(m) Sigma(chi not equal chi 0) vertical bar Sigma(a+b)(x=a+1) chi(x)vertical bar(4) << 8(nu(m))tau(m)(log m)(3)(log log m)(7) B-2. (C) 2009 Elsevier Inc. All rights reserved.