1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Number Theory Series Problem

  1. Aug 18, 2011 #1
    First note that the operation * denotes the Dirichlet product, and µ denotes the Möbius function.

    Ok so here is the problem:

    Let f(x) be defined for all rational x in 0≤x≤1 and let [tex]F(n)=\sum_{k=1}^nf(\frac{k}{n})\;\;\;\;\; and \;\;\;\;\;\; G(n)=\sum_{k=1,\;(k,n)=1}^nf(\frac{k}{n}).[/tex]

    Prove that G=µ * F.


    Formally here is what I've got: Set [tex]n=p_{1}^{e_1}...p_t^{e_t}.[/tex] than
    [tex]\mu * F = \sum_{d|n}[\;\mu (d)\sum_{k=1}^{\frac{n}{d}}f(\frac{dk}{n})\;][/tex][tex]=\sum_{k=1}^{n}f(\frac{k}{n})-[\;\sum_{k=1}^{\frac{n}{p_1}}f(\frac{p_1k}{n})+...+ \sum_{k=1}^{\frac{n}{p_t}}f(\frac{p_tk}{n})\;]+\sum_{k=1,\;i≠j}^{\frac{n}{p_ip_j}}f(\frac{p_ip_jk}{n})-...+f(\frac{n}{n}).[/tex]

    So this is probably a lot to take in but I'm pretty sure it's correct, just remember that the Möbius function is zero whenever d contains multiple copies of a prime in its factorization and thus many terms get zeroed out. Now, after working with some toy examples, the way I see this equaling G is that the initial series, which is just F, gets all its terms satisfying (k,n)>1 canceled out by terms coming from those series inside the brackets. But there are some terms in those series which have doubled up, tripled up, etc. and those terms will be canceled out by the later series(plural) which range over i,j and i,j,l and so on and so forth depending on the size of n. Now! The question is how the hell can I show this rigorously? It's just such a mess that there must be some way of simplifying the situation but I can't see it. Thanks.
     
    Last edited: Aug 18, 2011
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Number Theory Series Problem
  1. Number theory problem (Replies: 2)

Loading...