Show and .
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 .
Very good!
It can be shown , where are the Euler Numbers.
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Also if anyone didn't follow the first step, .
The second equality comes from the fact that , where
Now observe is completely multiplicative so we can transform into its Euler Product representaion.
Sort of... it's not the but rather a Dirichlet character.
The link I provided goes in depth and has tables for you to see, but this particular is the only non-principle character modulo .