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, $$\sum _{j=0}^{\infty }\frac{\alpha _{j}}{(j+1)^{s}}$$ ∑ j = 0 ∞ α j ( j + 1 ) s , $$\alpha _{0}=1$$ α 0 = 1 , of a class, say $${\mathcal {P}}$$ P , that contains in particular the series that define the Riemann zeta and the Dirichlet eta functions, there exists $$\lim _{n\rightarrow \infty }\rho _{n}/n$$ lim n → ∞ ρ n / n , where the $$\rho _{n}$$ ρ n ’s are the Henry lower bounds of the partial sums of the given Dirichlet series, $$P_{n}(s)=\sum _{j=0}^{n-1}\frac{\alpha _{j}}{(j+1)^{s}}$$ P n ( s ) = ∑ j = 0 n - 1 α j ( j + 1 ) s , $$n>2$$ n > 2 . Likewise it is given an estimate of the above limit. For the series of $${\mathcal {P}}$$ P having positive coefficients it is shown the existence of the $$\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n$$ lim n → ∞ a P n ( s ) / n , where the $$a_{P_{n}(s)}$$ a P n ( s ) ’s are the lowest bounds of the real parts of the zeros of the partial sums. Furthermore it has been proved that $$\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n=\lim _{n\rightarrow \infty }\rho _{n}/n$$ lim n → ∞ a P n ( s ) / n = lim n → ∞ ρ n / n .