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

Alex23
Member
Registered: 2012-01-31
Posts: 19

Crazy zeta function - infinity equals negative number!?

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. roll

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,270

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

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


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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

Alex23
Member
Registered: 2012-01-31
Posts: 19

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

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)

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,270

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

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.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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

Stangerzv
Member
Registered: 2012-01-30
Posts: 173

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

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.

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

Alex23
Member
Registered: 2012-01-31
Posts: 19

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

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.

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

Helping hand
Guest

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

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

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