Re: Characteristic function of exact divisors



"Gianfranco Oldani" <gf_oldani@xxxxxxxxxxx> schrieb im Newsbeitrag
news:8190245.1171027885239.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
let d be an integer divisor of the integer n. We say that d is an exact
divisor of n if gcd(d, n/d) = 1 i.e. if d and n/d are relatively prime. My
question is related to the characteristic function of the exact divisors
F(d,n) such that:

F(d,n) = 1 if d exact divisor of n, 0 else.
(d exact divisor also written as d||n)

Note that F(ab,nm) = F(a,n)F(b,m) if (n,m)=1, a ||n and b||m.

Is there a way to have an arithmetical expression of F(d,n) involving
well known arithmetical functions.

Thanks for help.


Yes:. Your function is

F(d,n) =T(n,d)-T(n,d^2),

whereby

T(n,m) = (1/m)*Sum_{k=1 to m} Cos(2*Pi*k*n/m)

is 1, if m a divisor of n, otherwise 0. The proof for the second
function T(n,m) you find under
http://de.wikipedia.org/wiki/Teilersumme#Teilersumme_als_endliche_Reihe
in the german wikipedia.

V.Thürmer




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