Crazy zeta function - infinity equals negative number!?

Alex23 · 2012-01-31 04:42:08

Alright, ζ(0) = 1 + 1 + 1 + ... = -1/2. Miracles do happen!

I want to comprehend what this means but it seems to me this just supersedes quantitative math.

Is there supposed to be a qualitative limited meaning to it?
Because if you forget the outcome is from zeta and just write 1 + 1 + 1 + ... = -1/2 this clearly is nonsense alone like that, as Hardy and Littlewood thought initially of it mailed by Ramanujan.

Math gurus enlighten me.

bobbym · 2012-01-31 06:45:59

Hi Alex23;

Welcome to the forum!

The zeta function is defined like this

only when the sum converges which it clearly does not for s=0. So the above definition does not hold. Look here for how an analytic continuation is used:

http://mathworld.wolfram.com/RiemannZetaFunction.html

Alex23 · 2012-02-02 07:27:48

I read the article. So the definition changes for that domain with gamma functions and what not.

Thanks for clearing up the definition does not hold for Re(s) < 1. In other forums people were just stating it is an unexpected result and we have to live with it and I was thinking WHAT??!, the equality is altogether wrong, meaning the sum over ones.

My question is how analytic continuation does not address a new function all together? Is it because the transformation is unique; that is the key?

Cheers!

Last edited by Alex23 (2012-02-02 07:28:30)

bobbym · 2012-02-02 07:32:13

Hi Alex23;

That is something I can not answer. I only have the tiniest bit of understanding of that page. Not enough to even comment.

Stangerzv · 2012-02-04 23:12:04

To others, infinity is a concept not a number but to me there is even and odd infinity. To them 1+infinity=infinity but to me 1+infinity>infinity. They got all those values like 1/12 or etc because they play with the infinity as they like. If you have a function of 1-1+1-1+1..infinity, if you use their concept you would get sometimes S=1/2 yet you know when it is even, S=0 and when it is odd, S=1 and this function alternates 0 & 1 to the infinity. I think people need to respect the infinity, otherwise we would be hay-wired. I do sometimes play with the infinity and I can proof that Zeta function

is not always true and converge to value 4.

Alex23 · 2012-02-05 08:57:22

That is a big statement. Are you sure 4 is correct?
That is a celebrated result and nowhere did I read it is an indeterminate series, but that it just converges to Euler's result.

Helping hand · 2012-12-15 02:08:16

Did you use Euler's definition that the zeta of 2n is equal to Bernoulli of 2n times 2pi to the 2n over 2 times 2n factorial times minus 1 to the n+1th power?
If you did then it did not work because it only works for non-zero positive even integers because the Bernoulli of 0 can either be -1/2 or 1/2

Math Is Fun Forum

#1 2012-01-31 04:42:08

Crazy zeta function - infinity equals negative number!?

#2 2012-01-31 06:45:59

Re: Crazy zeta function - infinity equals negative number!?

#3 2012-02-02 07:27:48

Re: Crazy zeta function - infinity equals negative number!?

#4 2012-02-02 07:32:13

Re: Crazy zeta function - infinity equals negative number!?

#5 2012-02-04 23:12:04

Re: Crazy zeta function - infinity equals negative number!?

#6 2012-02-05 08:57:22

Re: Crazy zeta function - infinity equals negative number!?

#7 2012-12-15 02:08:16

Re: Crazy zeta function - infinity equals negative number!?

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