3
$\begingroup$

How to prove or justify the following:

$$ f(g)= \frac1{1-g^2} \prod_{k=1}^{\infty} \left(\frac{\sin(\pi \frac gk)}{\pi \frac gk} \cdot \frac 1{1-\frac{g^2}{k^2}}\right). $$ The above statement can illustrate the following facts:

(1) if $f(g) = 0$, then $g$ is composite number
(2) If $f(g)$ is not equal to $0$, the $g$ is prime number
(3) if $f(1+g) = 0$, then $g$ is prime.

Please justify.

  • 0
    @martini! Thank you so much for your editing work.2012-11-16
  • 2
    Note that $1/(1-g^2)$ is redundant2012-11-16
  • 1
    I am confused. What is $f$? It is given by the formula, then the formula is true by definition. And if not, how is it defined?2012-11-16
  • 0
    f is given by the formula. of course, formula is true by definition.2012-11-16
  • 0
    Oh, so it's the facts *after* the formula that you want proved or justified? That was not clear at all.2012-11-16
  • 2
    For me it seems that in this form, (3) contradicts (1) and (2).2012-11-16
  • 0
    I assume you meant $k>1,~$ since otherwise the first term of the infinite product is trivially zero. It all boils down to proving that $~\displaystyle\prod_{k=2}^{g-1}\sin\bigg(\pi~\frac gk\bigg)=0~$ for composite values of *g*, which is trivial, since the fraction $g/k$ is an integer for all instances of $k\mid g$.2018-11-26

0 Answers 0