On the equation n1n2=n3n4n1n2=n3n4 and mean value of character sums

For B⩾1B⩾1 let N(B)N(B) denote the number of solutions of the equation n1n2=n3n4n1n2=n3n4 with 1⩽ni⩽B1⩽ni⩽B. For a prime p let χ   denote a multiplicative character (modp). In this paper, we obtainN(B)=12π2B2logB+CB2+O(B547416+ε);1p−1∑χ≠χ0|∑x=1Bχ(x)|4=12π2B2logB+(C−B2p)B2+O(B547416+ε),B⩽p, for some constant C   and any ε>0ε>0, which improved the corresponding error terms O(B1913log713B) obtained by A. Ayyad, T. Cochrane and Z.Y. Zheng.