I consider

which is similar to zeta function .

I find that it can be ' factorised ' in this form :

The reason is when we expand the product from RHS we obtain something like :

For a number if it is , no matter how many primes it contains , the number must be even , and for , we conclude that the number of primes as factors must be odd , so we have this formula (1) .

Together with

We obtain , by division :

Also , we have

so we obtain

I don't know the value of but from your writing , it should be and if I remember correctly , we should have

, it looks ok to me ...

Remarks : I've read a book by Euler , called Introduction to Analysis of the Infinite , in this book he really introduced a method of evaluating ( of course , in the mean time , he evaluated , too ) but i forgot this method , recently i got another method using Fourier Analysis , I guess you experts are using this skilfully .