We can start from the explicit expression of ...

(1)

... from which we derive immediately...

(2)

Now we remember the well known 'Euler's product' written in the 'not very usual form' ...

(3)

... from which, setting we derive immediately...

(4)

The (4) tells us that for any we have an integer so that is...

(5)

Combining (2) and (5) we conclude that for any it exist a k so that setting...

(6)

...is...

(7)

Kind regards