Ramanujan identities and Euler products for a type of Dirichlet series

has an Euler product and gave an explicit formulation for the Euler product. In this paper we develop the theory of binary quadratic forms in order to determine the Euler product for ∑∞ n=1 a(n)n −s, and other similarly defined Dirichlet series, in a completely elementary and natural manner. Let N, Z, R and C be the sets of natural numbers, integers, real numbers and complex numbers respectively. A nonsquare integer d with d ≡ 0, 1 (mod 4) is called a discriminant. The conductor of the discriminant d is the largest positive integer f = f(d) such that d/f ≡ 0, 1 (mod 4). As usual we set w(d) = 1, 2, 4, 6 according as d > 0, d < −4, d = −4 or d = −3. For integers a, b and c, we use (a, b, c) to denote the integral, binary quadratic form ax + bxy + cy. The form (a, b, c) is said to be primitive if gcd(a, b, c) = 1. The discriminant of the form (a, b, c) is the integer d = b − 4ac. If d < 0, we only consider positive definite forms, that is, forms (a, b, c) with a > 0 and c > 0. Two forms (a, b, c) and (a′, b′, c′) are equivalent ((a, b, c) ∼ (a′, b′, c′)) if there exist integers α, β, γ and δ with αδ − βγ = 1 such that the substitution x = αX+βY, y = γX+δY transforms (a, b, c) to (a′, b′, c′). It is known that (a, b, c) ∼ (c,−b, a) and for k ∈ Z that (a, b, c) ∼ (a, 2ak + b, ak + bk + c). We denote the equivalence class of (a, b, c) by [a, b, c]. The equivalence classes of primitive, integral, binary quadratic forms