Show and .

Printable View

- July 15th 2010, 05:11 PMchiph588@Euler Product
Show and .

- July 15th 2010, 10:50 PMsimplependulum

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 (Happy) . - July 16th 2010, 12:42 PMchiph588@
Very good! (Clapping)

It can be shown , where are the Euler Numbers.

-------

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. - July 16th 2010, 08:04 PMsimplependulum
Oh , is the Dirichlet function ? I first saw it in the later part of a book about elementary number theory .

- July 16th 2010, 09:01 PMchiph588@
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 .