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Limit of the Euler totient function

  1. Apr 22, 2013 #1
    My question is relatively breif: is it true that

    [tex]\displaystyle \lim_{n \rightarrow \infty}(\varphi(n))=\lim_{n \rightarrow \infty}(n) \cdot \prod_{i=1}^{\infty}(1-\frac{1}{p_i})[/tex]
    Where [itex]p[/itex] is prime? Pehaps [itex]\varphi(n)[/itex] is too discontinuous to take the limit of, but it would seem that as it increases to infinity the function should tend to infinity, with fewer anomalies.

    If this were true,

    [tex]\displaystyle \zeta(1)=\frac{1}{ \prod_{i=1}^{\infty}(1-\frac{1}{p_i})}=\frac{1}{\lim_{n \rightarrow \infty}(\frac{\varphi(n)}{n})}=\lim_{n \rightarrow \infty}(\frac{n}{\varphi(n)})[/tex]
     
  2. jcsd
  3. May 3, 2013 #2
    Let ##\mathfrak{P}_n## be the set of all distinct prime divisors of a number n.
    Consider that ##\displaystyle \varphi(n) = n \prod_{i=1}^{\sharp\mathfrak{P}_n}\left[1-\frac{1}{p_i}\right]##, where ##\sharp\mathfrak{P}_n## is the cardinality of ##\mathfrak{P}_n## and ##p_i## is the ith element of ##\mathfrak{P}_n##. As n increases, its number of prime factors tends to increase, but this trend is in no way strictly true for individual numbers. An example of a relatively large number that does not have a large number of prime factors is 87178291199, which has only one prime factor. :tongue:

    Thus, using basic properties of limits, your formula should be correct.
     
  4. May 4, 2013 #3
    The problem I has was that if particular numbers are 'discontinuous' from the general trend, you can't take the limit, even if the general trend tends to infinity.
     
  5. May 4, 2013 #4
    Consider a function ##f: \mathbb{R}\rightarrow\mathbb{R}\cup\left\{Tootsiepop\right\}##, where, for ##x\in\mathbb{R}##, ##f(x)=\left\{\begin{array} , x , x\neq2 \\ Tootsiepop , x=2 \end{array}\right.##

    As x approaches 2, f(x) approaches 2. However, f(2)=Tootsiepop.

    This is not a continuous function, but the limit as x approaches 2 is defined. So, I don't understand what you mean...
     
  6. May 4, 2013 #5
    The example cleared up a lot, thanks. I've little formal experience with limits.
     
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