Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

摘要

For empty set not equal P subset of P, be the arithmetic subderivative function with respect to P on Z(+), let zeta D-p be the function defined by the Dirichlet series of D-p, and let sigma(DP) denote its abscissa of convergence. Under certain assumptions concerning s and P, we present asymptotic formulas for the partial sums of zeta(DP)(s) and show that sigma(DP) = 2. We also express zeta(DP)(s), s > 2, using the Riemann zeta function.