Sum of Positive Integers and the Riemann Zeta Function (1 Viewer)

frog1944 · Jul 31, 2016

Hi,

I've seen and heard a lot about how the sum of all positive integers to infinity is -1/12 (This numberphile video was where I originally found it: https://youtu.be/w-I6XTVZXww). Is this true? I heard that somehow the Riemann Zeta function confirms it and they gave a proof with the Zeta function and another without it.

So is this true?
Or only under some circumstances?

Thanks

Nailgun · Jul 31, 2016

Paradoxica posted this awhile ago

Paradoxica said:
$$\noindent Your fear is somewhat correct, as I will now pull out the Riemann Zeta Function, which is used in the proof of $ \sum_{k=1}^\infty k = -\frac{1}{12}$

Correct, although I don't actually fully understand regularisation yet.

$\zeta(s) = \sum_{k=1}^\infty k^{-s}$

$\noindent The derivative of the zeta function evaluated at $s = 0$ is $-\frac{1}{2}\log(2\pi). $ I am still scouring the internet for a proof of this fact, as I cannot seem to find a proof of it anywhere.$

$$\noindent Differentiating with respect to $s$, we have: $\zeta'(s) = \sum_{k=1}^\infty (-\log{k}) k^{-s} \Rightarrow \Rightarrow \Rightarrow \Rightarrow \zeta'(0) = -\sum_{k=1}^\infty \log{k}$

$$\noindent Combining the above, we have: $ -\sum_{k=1}^\infty \log{k} = -\log{\sqrt{2\pi}} \Rightarrow \Rightarrow \Rightarrow \Rightarrow \Rightarrow \Rightarrow \Rightarrow \Rightarrow \sum_{k=1}^\infty \log{k} = \log{\sqrt{2\pi}}$

$\log{1} + \log{2} + \log{3} + \dots + \log{\infty} = \log{\infty !} = \log{\sqrt{2\pi}}$

$\therefore \infty ! = \sqrt{2\pi}}$

seanieg89 · Jul 31, 2016

To be able to say that the statement is "true" we need to come up with a way of making sense of a divergent series like this one. This is the real issue of the question, because there are multiple ways of doing this. The most obvious such method makes use of the analytic continuation of the Dirichlet series that defines the zeta function. (Others proceed by real variable methods, but many of these approaches are intertwined with the zeta function anyway.)

Some exposition can be found at the start of https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

I stress though, that this comes down to a matter of definitions. That sum is equal to -1/12 under a suitable definition of what it means to take a sum. The sum is still a divergent one in the usual sense. This is why weird phenomenon like the sum of a bunch of positive integers being negative can occur, you cannot recklessly carry across the intuition that comes from series that are convergent in the usual sense to this notion of summation.

frog1944 · Jul 31, 2016

Ok, thank you very much. That is very interesting about the definitions, I never thought of it that way. I will have a read of the blog post

seanieg89 · Jul 31, 2016

Yep, that's the danger of colloquial statements like:

"The sum of the positive integers is -1/12".

They are catchy titles for articles/videos/blogposts but are very imprecise and misleading.

Sum of Positive Integers and the Riemann Zeta Function (1 Viewer)

frog1944

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Nailgun

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seanieg89

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frog1944

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seanieg89

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