Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#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

Offline

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,678

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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)

Offline

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,678

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

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

Stangerzv
Member
Registered: 2012-01-30
Posts: 194

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.

Offline

#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.

Offline

#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

Board footer

Powered by FluxBB