### 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).

(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

#### Full Text:

PDFDOI: http://dx.doi.org/10.5269/bspm.v38i2.34878

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