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).