Abelian and Tauberian results on Dirichlet series
Consider the sequence \(\{a_n\}\) of complex numbers and an asymptotic formula \[ \sum_{n\leq x} a_n= xP(\log x)+ O(x^\beta L(x)),\quad P(y)= \sum^k_{m=0} p_m y^m, \] where \(L\) is a slowly oscillating function. Conditions are supplied under which the coefficients, \(C_r\), of the Laurent expansion \[ \sum^\infty_{n=1} {a_n\over n^s}= \sum^k_{m=0} {p_m m!s\over (s-1)^{m+1}}+ \sum^\infty_{r=0} {(-1)^r C_r\over r!} (s- 1)^r \] can be obtained. Consequently, an asymptotic formula is obtained for \[ \sum_{n\leq x} {a_n\over n}\log^r n. \] This provides a generalization of the results of \textit{W. E. Briggs} [Mathematika 9, 49--53 (1962; Zbl 0166.05401)]. Several examples for specific arithmetic functions are included.
scientific article; zbMATH DE number 5009751