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Numberphile v. Math: the truth about 1+2+3+...=-1/12

Mathologer4 HN recommenders · recommended in 4 months, 2020–2023
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explainer · intermediate · evergreen · 2020–2023 on HN

Mathologer examines the infamous Numberphile claim that 1+2+3+... equals -1/12, explaining what is wrong with the calculation and how it connects to real math like the zeta function and analytic continuation.

HN points to it as the in-depth rebuttal to the Numberphile video that explains why the 'proof' is wrong yet the result is meaningful.

Recreational & visual mathPure mathematics#-1/12#zetafunction#analyticcontinuation#ramanujan

7 comments from Hacker News

@jonnytran2022-07 · in “Ask HN: YouTube Channels for the Intellectually Curious”

I know you're asking about Computerphile, but this video explains how one of the Numberphile videos is wrong: ▶ this video

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@crvdgc2021-10 · in “Conterintuitive facts in mathematics, CS, and physics”

I find a response video to it more enlightening. It's often found as the number one related video. ▶ this video

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@cyphar2020-11 · in “1.5 is the midpoint between 0 and infinity in Ruby”

That proof (and the more general statement) is not correct because it makes assumptions about infinite summation of divergent series that are not true. There is a link between the sum of natural numbers -- namely, the analytic continuation of the Reimann Zeta function has Z(-1) = -1/12. But the Zeta function is only defined to equal the sum of inverse powers for Re(z) > 1. And under a specific definition of summation (Ramanujan summation) you can say that "the sum is equal to -1/12" but that isn't the same as normal summation. Mathologer did a fairly in-depth video[1] into why this proof is wrong and what the actual link is between the infinite series and -1/12. The upshot is that even if you use more complicated definitions of summation, you cannot define sums of the kind 1+2+3+... to equal a finite number. If you assume the infinite sum of 1+2+3+... converges to a finite number you can easily prove a contradictory statement using the same summation properties assumed by that proof. Namely: S = 1 + 2 + 3 + ... = -1/12 S2 = S-S = 0 = 1 + 2 + 3 + ... - 1 - 2 - 3 - ... = 0 = 1 + 2 + 3 + ... - 1 - 2 - ... = 0 S2 = 1 + 1 + 1 + 1 + ... = 0 S3 = S2-S2 = 0 = (1-1) + (1-1) + ... = 1 - 1 + 1 - 1 + ... = 0 But we "derived" in the original proof that S3 is equal to 1/2, which is a contradiction. (You could derive S3 is 1/2 from S but that's what the article does in reverse.) [1]: ▶ this video

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@stan_rogers2020-02 · in “Ramanujan Surprises Again (2015)”

Burkard Polster's (Mathologer) videos on the subject are probably going to be more useful than Numberphile. Numberphile merely presents a trick; Mathologer points out both that it's patent nonsense [0] and that it's useful patent nonsense [1]. [0] ▶ this video [1] link

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@OldGuyInTheClub2023-09 · in “A Duodecade of Numberphile”

Yes, I kept waiting for them to "fess up" at the end but they didn't. Mathologer called them out on it. ▶ this video

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