Converegence of a series leading to an analogue of Ramanujan's assertion on squarefree integers

Abstract

Let d be a squarefree integer. We prove that (i) Pn μ(n) n d(n′) converges to zero, where n′ is the product of prime divisors of n with ( d n ) = +1. We use the Prime Number Theorem. (ii) Q( d p )=+1(1 − 1 ps ) is not analytic at s=1, nor is Q( d p )=−1(1 − 1 ps ) . (iii) The convergence (i) leads to a proof that asymptotically half the squarefree ideals have an even number of prime ideal factors (analogue of Ramanujan’s assertion).

Keywords

• Dirichlet series
• Prime Number Theorem