This paper discusses the Dirichlet's L-function L(s,χ) for a character χ mod k, and we prove a refinement of the error term using the result on Hurwitz Zeta function, proved by the author in an earlier paper.
Let Δ(x)=∑n≤xa(n)−∑6j=1cjx1/j denote the error term in the abelian group problem. Using zeta-function methods it is proved that ∫X1Δ2(x)dx<<X39/29log2X
where the exponent 39/29=1.344827… is close to the best possible exponent 4/3 in this problem.