Open AccessConveregence of a series leading to an analogue of Ramanujan's assertion on squarefree integers
Author(s) -
G. Sudhaamsh Mohan Reddy,
S. Srinivas Rau,
B. Uma
Publication year - 2018
Publication title -
boletim da sociedade paranaense de matemática
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.347
H-Index - 15
eISSN - 2175-1188
pISSN - 0037-8712
DOI - 10.5269/bspm.v38i2.34878
Subject(s) - mathematics , square free integer , ramanujan's sum , assertion , combinatorics , prime (order theory) , series (stratigraphy) , discrete mathematics , computer science , programming language , paleontology , biology
Let d be a squarefree integer. We prove that(i) Pnμ(n)nd(n′) converges to zero, where n′ is the product of prime divisors of nwith ( dn ) = +1. We use the Prime Number Theorem.(ii) Q( dp )=+1(1 −1ps ) is not analytic at s=1, nor is Q( dp )=−1(1 −1ps ) .(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).