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 $k\ge 0$, 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.

arXiv comment24 pages