This paper discusses the Dirichlet's $L$-function $L(s,\chi)$ for a character $\chi$ 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 $\Delta(x)=\sum_{n\leq x}a(n)-\sum_{j=1}^6 c_jx^{1/j}$ denote the error term in the abelian group problem. Using zeta-function methods it is proved that
$$\int_1^X\Delta^2(x)\,dx~<\!\!<~ X^{39/29} \log^2X$$
where the exponent $39/29=1.344827\ldots$ is close to the best possible exponent $4/3$ in this problem.