On the converse of Mertens' theorem

Abstract

Let (λm), (µn) (m, n = 0, 1, 2,…) satisfyrespectively. Let vp (p = 0, 1, 2, …) be the sequence (λm+µn) arranged in ascending order, equal sums λm+µn being considered as giving just one vp Then for given formal series Σam, Σbn the formal series C = Σ cp whereis called the general Dirichlet product of Σamand Σbn (see Hardy (2), p. 239). When λn = µn = n we have the Cauchy product. In the case λn = logm, µn = logn (m, n = 1, 2,…) we have vp =log p(p = 1, 2, …)and it is natural to call C the ordinary Dirichlet product.