On the lower bounds of the partial sums of a Dirichlet series

Abstract

In this paper it is shown that for the ordinary Dirichlet series, ∑∞j=0αj(j+1)s, α0=1, of a class, say P, that contains in particular the series that define the Riemann zeta and the Dirichlet eta functions, there exists limn→∞ρn/n, where the ρn’s are the Henry lower bounds of the partial sums of the given Dirichlet series, Pn(s)=∑n−1j=0αj(j+1)s, n>2. Likewise it is given an estimate of the above limit. For the series of P having positive coefficients it is shown the existence of the limn→∞aPn(s)/n, where the aPn(s)’s are the lowest bounds of the real parts of the zeros of the partial sums. Furthermore it has been proved that limn→∞aPn(s)/n=limn→∞ρn/n.