Show and .
Show and .
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) .
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 (Happy) .
Oh , is the Dirichlet function ? I first saw it in the later part of a book about elementary number theory .
The link I provided goes in depth and has tables for you to see, but this particular is the only non-principle character modulo .