## Euler's Zeta function equality and the Fall from Infinity

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• 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 +
Message 1 of 4 , Jun 4, 2003
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.

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
• ... The zeta function is convergent for all complex numbers except for n=1. Satoshi Tomabechi
Message 2 of 4 , Jun 5, 2003
"Mark Underwood" <marku606@...> wrote:

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

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

Satoshi Tomabechi
• In a message dated 05/06/03 06:26:35 GMT Daylight Time, marku606@yahoo.ca ... Why so? tan(theta) falls from infinite to finite for an infinitely small
Message 3 of 4 , Jun 5, 2003
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
> with (I presume) an infinitely small increment of the n exponent.
>

Why so?
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 ... Infinitely or finitely? Consider 1/e. As e- 0 the expression has a value, at e=0 it
Message 4 of 4 , Jun 5, 2003
'> 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/