# Some details for Zeta functions

Here I’m including proofs and derivations for some technical results used in Ogus’ discussion of Zeta functions.

For positive integers with e fixed, consider the function

Then where is the set of all th roots of unity (this is the first result). This makes intuitive sense, when n is a multiple of e all the terms in the sum are 1, and in all other cases the roots are permuted and you sum them to get zero. here’s Ogus’ formal proof:

For brevity a product or sum over means . So

and

The second equality follows because the the leading coefficient should be +1 and all are roots of the rhs. Note that has the property that . Applying this to

and comparing sides yeilds

Expanding as power series gives

This gives the first result.

Here’s is the next result. Let be a finite number of complex numbers and set

Adding zeros if necessary, it can be assumed that there are an equal number of a_i and b_i. The second result is, as a formal power series,

where . This is just some manipulations:

Or equivalently, so integrating gives

and exponentiation gives the second result.

## About this entry

You’re currently reading “Some details for Zeta functions,” an entry on Math Meandering

- Published:
- August 18, 2009 / 2:03 pm

- Category:
- Ogus Excerpts

- Tags:

## No comments yet

Jump to comment form | comment rss [?] | trackback uri [?]