scott <sco### [at] scottcom> wrote:
> Doesn't that come under a different category of just being a trick/hoax 
> though (a bit like all the 1=2 type "proofs")? As opposed to this 
> assuming 1+2+3+...=-1/12 thing is actually useful in other areas of 
> maths and science.

I'm still not sure how valid it is to use "1+2+3+..." here, even when
talking about physics.

The so-called "Euler zeta function" is the infinite sum, where n goes
from 1 to infinity, of 1/n^s, where 's' is a real number. This infinite
sum converges to a finite value for any value of s > 1. For any value
of s <= 1 the sum converges to infinity (and thus is undefined).

Bernhard Riemann had an epiphany about said function when he was
studying it (something about it being related to the density of
prime numbers), and he extended it for all complex values of s.

The infinite sum still converges to a finite value when the real
part of s is > 1 (the imaginary part can be anything), and to infinity
when the real part is <= 1 (and thus is undefined.)

There is a way, however, to extend such functions to cover the entire
complex plane in such a manner that the result is still the same for
all Real(s)>1, but defined for all the remaining complex values of s
as well (except for the single singularity at s=1+0i, which remains
undefined).

This so-called analytical continuation of the Euler zeta function is
the so-called Riemann zeta function. Said function gives the exact
same values as the former for all Real(s)>1. However, the function
is rather different from the much simpler Euler zeta function. It's
not the same function.

It turns out that the Riemann zeta function gives a value of -1/12
when s = -1. If you were to plug s = -1 into the Euler zeta function,
you would get the infinite sum 1+2+3+4+5... (try it to see.)

However, the Euler zeta function is *not* the Riemann zeta function.
They give different results for all Real(s) <= 1. When you plug s=-1
into the Riemann zeta function, you are *not* getting 1+2+3+4+5+...
You are getting something completely different (something that results
in -1/12).

Why they are somehow considered "equal", I don't understand. (I'm not
a mathematician.)

-- 
                                                          - Warp

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