- Hi all

Checking out the Prime Pages, I was astonished to see a relation that

Euler came up with, concerning his Zeta function:

Z(n) = 1/1^n + 1/2^n + 1/3^n + 1/4^n + 1/5^n + 1/6^n ....

= 2^n/(2^n - 1) x 3^n/(3^n - 1) x 5^n/(5^n - 1) x

7^n/(7^n - 1) x 11^n/(11^n - 1) x ...

I find this relationship of the natural numbers to the primes to be

incredible and mystifying.

Recently someone asked about a Zeta like function diverging or

converging, which got me wondering about something: If the Zeta

function is divergent for n = 1 (becomes infinite), and if it

converges to a finite number when n = 2 ( Z(2) incredibly equals

(pi^2)/6 ), that must imply that there is a value for n between 1

and 2 as n increases at which the series somehow falls from an

infinite sum to a finite sum. What value of n would that be, anyone?

It strains my brains that the sum could fall from infinite to finite

with (I presume) an infinitely small increment of the n exponent.

Mark - "Mark Underwood" <marku606@...> wrote:

> Z(n) = 1/1^n + 1/2^n + 1/3^n + 1/4^n + 1/5^n + 1/6^n ....

The zeta function is convergent for all complex numbers except for n=1.

>

> (pi^2)/6 ), that must imply that there is a value for n between 1

> and 2 as n increases at which the series somehow falls from an

> infinite sum to a finite sum. What value of n would that be, anyone?

Satoshi Tomabechi - In a message dated 05/06/03 06:26:35 GMT Daylight Time, marku606@...

writes:

> It strains my brains that the sum could fall from infinite to finite

Why so?

> with (I presume) an infinitely small increment of the n exponent.

>

tan(theta) falls from infinite to finite for an infinitely small increment of

theta from pi/2...

Mike Oakes

[Non-text portions of this message have been removed] - '> It strains my brains that the sum could fall from infinite to finite
> with (I presume) an infinitely small increment of the n exponent.

Infinitely or finitely?

> '

Consider 1/e. As e->0 the expression has a value, at e=0 it is undefined.

This is basic continuity theory.

The behaviour of zeta(s) around s=1 is analysed at;

http://numbers.computation.free.fr/Constants/Miscellaneous/zeta.html

Jon Perry

perry@...

http://www.users.globalnet.co.uk/~perry/maths/

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