Note on Absolutely Convergent Dirichlet Series

is it true that \f(s)\ s^k>0 for cr^O implies that (/(s))-1 is also of the form (1)? In this note, an affirmative answer is supplied.3 Let P be the semigroup of positive integers under multiplication, and let h(P) be the class of all complex functions a on P, a= {an}»~i, for which ||a|| = y^°-i \an\ is finite. We obtain a commutative Banach algebra by defining (aa)n = aan for complex a, (a+b)n = an+bn, and the product a*b (convolution) by (a*b)n= 22jk=najbk. The algebra h(P) has as its unit e the function such that ex= 1 and en = 0 if n> 1. The theory of /i-algebras of commutative semigroups is developed in [4] and [5]. The present note is based on the observation that the Dirichlet series (1) is the Fourier transform of the element aEh(P), defined on a very small subspace of the space of all maximal ideals of k(P).