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

G. Sudhaamsh Mohan Reddy, S Srinivas Rau, B. Uma

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

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



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