Note on Type L-Functions of Euler Product of Second Degree

Abstract

In this paper we are concerned with a special case of Euler functions. We shall study the Euler product of degree two (Type of L-Euler functions) and give some results. More precisely, we shall deal with some Dirichlet series associated with a class of arithmetic {an} under the condition that apapk = apk+1 + pαapk−1, provided p is prime, k≥1, and α is a fixed complex number. We will demonstrate that there is an Euler’s product for the Dirichlet series Σnan/ns. This result is important in analysis, especially in analytic number theory.