Riesz summability of Dirichlet series generating holomorphic functions of finite order

Abstract

Given a frequency $\lambda$, we study the Riesz summability of $\lambda$-Dirichlet series $\sum_{n=1}^\infty a_n e^{-\lambda_n s}$ generating holomorphic functions of finite order. We present a new separation condition on the frequency $\lambda$ ensuring that, for any $k \geq 0$, each $\lambda$-Dirichlet series that is somewhere Riesz summable of some order and admits a holomorphic extension $f$ to the right half-plane $\mathbb{C}_0$ satisfying $f(s) = O(|s|^k)$ as $|s| \to \infty$ on $\mathbb{C}_0$, is in fact Riesz summable of order $k$ on $\mathbb{C}_0$. This extends Bohr's theorem, which corresponds to the case $k = 0$. Our work improves a recent result of Defant and Schoolmann, who showed the above property under Landau's condition (LC). Along the way, we also establish novel bounds on the coefficients of such $\lambda$-Dirichlet series and, under a mild condition on the frequency $\lambda$, show that they are optimal.