18 Feb
2012
18 Feb
'12
3:43 p.m.
Fred: Did you add anything here? Asking only because I have come to expect gems whenever you chime in. On 2/18/2012 4:33 PM, Fred lunnon wrote:
On 2/18/12, Michael Reid<reid@gauss.math.ucf.edu> wrote:
Letting mu be the Moebius function, and nu be the Mertens function defined as
nu(n) = SUM(k = 1..n, mu(k))
Then I notice
[1] SUM(k = 1..n, mu(k)*[n/k]) = 1
[2] SUM(k = 1..n, nu([n/k])) = 1
where [x] is the floor of x. I wondered if there was any simple relationship linking [1] and [2], or perhaps some relationship to the Sure, write the second sum as a double sum, using the definition of nu(-) , and then switch the order of summation.
Moebius transform
a(n) = SUM(d|n; b(d))<=> b(n) = SUM(d|n; mu(n/d)*a(d)).