On the Dirichiet product of Cesàro-summable series

Abstract

Segal(1) in his paper ‘Summability by Dirichlet convolutions’ makes direct use (in the proofs of Theorems 4 and 6) of a result on the Dirichlet product of two series which may be stated as follows: is (C, k)-summable to the sum α (k≥0) and is (C, l)-summable to the sum β, (l ≥ 0), then the Dirichlet productis (C, k + l + 1)-summable to the sum αβ.’