Does -1/12 Protect Us From Infinity? - Numberphile

Source: Does -1/12 Protect Us From Infinity? - Numberphile, Numberphile, 21:20, uploaded 2024-02-16.

Tony Padilla returns to the sum $1+2+3+4+\ldots$ ten years after Numberphile first attached the value $-1/12$ to it. The old video drew a fair amount of anger because the ordinary partial sums grow without bound. Padilla agrees that the usual sum diverges. His new question concerns the way we approach infinity: a sharp cutoff produces the familiar divergence, whilst smooth regulators can expose a finite term that remains fixed across a whole family of choices.

The old controversy and the ordinary partial sum

Brady Haran introduces the subject as a return to a dark place. Padilla says that the earlier video told a story about the sum of the natural numbers and used an equal sign before the explanation had earned it. The result upset people, although it also made many viewers interested in divergent series. About a year and a half before filming, a tweet annoyed Padilla enough to make him calculate a rebuttal. The calculation led to a research project with his student Robert Smith and to the paper that accompanies this video.

The usual way to approach an infinite series is to stop at a finite term and then move the stopping point outwards. For the natural numbers, the partial sum is

As $N$ grows, this expression grows like $N^2$. Sending $N$ to infinity therefore produces a divergence. The calculation contains no trace of $-1/12$, which is why the result from the older video looks impossible when it is read as an ordinary equality.

Padilla says that Terence Tao helped him see what the partial sum does to each term. It multiplies the infinite series by a step function that equals one up to $N$ and zero after $N$. The step changes value at one point, so the procedure imposes a sudden transition on the series. The partial sum is a legitimate method for a convergent series. For this divergent series, it is one choice of cutoff with a particularly abrupt edge.

Smooth cutoffs and the finite term

Padilla replaces the step with a waiting function that starts at one and falls towards zero around the scale $N$. The regularised series becomes

where $\eta(0)=1$ and $\eta(x)$ decreases smoothly as $x$ grows. The first example uses $\eta(x)=e^{-x}$. The resulting sum still has a quadratic divergence, so smoothing alone does not make the whole expression finite. Its large-$N$ behaviour has the form

The coefficient $C_1[\eta]$ depends on the waiting function. Padilla describes it as the integral of $x\eta(x)$, the first Mellin transform of the regulator. The quadratic part therefore changes when the regulator changes. The finite term $-1/12$ remains the same. In the ordinary partial sum, the sharp step hides that term beneath its divergence. A smooth transition separates the regulator-dependent growth from the finite piece.

This is the part that makes the result precise. The regularised expression has a finite term equal to $-1/12$ whilst the original series remains divergent in the ordinary Cauchy sense. The value comes from a summation procedure with extra structure. It does not turn the sequence of partial sums into a convergent sequence.

Enhanced regulators

Padilla then chooses a smooth waiting function whose first Mellin transform vanishes. The paper calls such a function an enhanced regulator. With $\eta(x)=e^{-x}\cos(x)$, the quadratic term disappears and

This route reaches the finite value without first producing a divergent term that someone has to discard. Padilla admits that he reverse-engineered the cosine factor after seeing the desired result. That admission matters because the regulator is a construction chosen for its properties. The finite value is mathematically controlled once the regulator has been specified, while the choice of regulator still needs a reason.

The paper gives a general family of smooth regulators with the same property. It also shows that the finite term comes from the Euler-Maclaurin expansion and agrees with the value obtained through zeta-function or Ramanujan-style regularisation. The video keeps the history of those methods in the background and stays with the picture of a sharp or smooth transition.

Padilla’s point is that the sharp cutoff is only one way to move through the terms of an infinite series. An infinite collection of smooth choices can reach the same finite term. The partial sum follows one particularly severe route because it keeps every term at full weight until a precise point and then removes all later terms at once. The existence of other routes does not make the ordinary route wrong. It changes which question the calculation answers.

From divergent series to quantum field theory

The pattern catches Padilla’s attention because quantum field theory also produces infinities that need a regulator. Loop integrals sum contributions over continuously varying momenta. Their power-law divergences depend on how the high-momentum part is controlled, whilst logarithmic terms have a more stable form across regularisation schemes.

Padilla and Smith introduce what they call $\eta$ regularisation. They multiply Euclidean loop integrals by a smooth function such as $\eta(|k|/\Lambda)$, where $|k|$ is the loop momentum and $\Lambda$ sets the cutoff scale. The Mellin transforms of the regulator control the power-law terms in the same way that $C_1[\eta]$ controls the quadratic term in the sum of natural numbers. An enhanced regulator can remove that power-law divergence.

The physical constraint comes from symmetry. In a gauge theory, Ward identities require the gauge-field amplitudes to remain transverse. A regulator that breaks those identities changes the structure the theory relies on. The paper tests its $\eta$ regularisation against the one-loop consistency conditions for non-abelian gauge theories coupled to Dirac fermions. It finds that the regulators that remove the quadratic divergence also satisfy the leading conditions needed to preserve gauge invariance. Different loop integrals can require related regulators, and the finite parts impose further constraints.

The connection is more specific than a visual resemblance between two calculations. The paper develops a regularisation scheme for loop integrals and finds that its enhanced regulators align with one-loop gauge consistency. It does not show that string theory explains the value $-1/12$, or that nature has selected one regulator. Padilla describes a possible string-theory connection because string amplitudes can soften high-energy behaviour, and the paper points towards Schwinger proper-time methods and future work. The deeper explanation remains open.

What infinity permits us to understand

The final exchange returns to ordinary intuition. Most people would say that adding all positive natural numbers gives infinity. Padilla says that no simple image can make $-1/12$ feel intuitive. Human experience takes place at finite scales, and even mathematicians learn to work with infinity through definitions rather than direct perception. He recalls the physicists who say that they have swept infinities under the rug whilst describing calculations in quantum field theory.

His preferred explanation is therefore procedural. The partial sums follow one sharp cutoff and diverge. Smooth regulators follow other paths. Some of those paths retain a universal finite term, and enhanced choices remove the regulator-dependent divergence as well. The finite result is special because it survives across a family of smooth constructions and because related regulator choices preserve the symmetries of a one-loop quantum field theory.

The video leaves the last claim at the right level of uncertainty. Padilla and Smith have found a connection between the regularisation of divergent series and gauge-invariant regularisation in quantum field theory. They do not yet know whether it reveals a deeper principle. The value $-1/12$ therefore remains a regularised finite term with a useful physical history, rather than the ordinary sum of the natural numbers.

Limits

The source is an accessible account of the paper’s idea, with equations shown on screen and several technical steps left implicit. The linked paper establishes the smoothed-asymptotic formula and the one-loop gauge-invariance result, whilst its conclusions describe higher loops, gravity and string theory as open directions. The video’s English caption track repeats a fragment of the opening during its final seconds, so that repeated fragment is excluded as a caption artefact.

The video description links Tony Padilla’s earlier explanatory response through an old University of Nottingham URL that currently returns a 404 page. Brady Haran’s linked blog preserves a copy of the response as a PDF, which explains the original video’s analytic-continuation argument and admits that the introductory manipulations were chosen for a general audience. That response supports the historical context of the controversy. It does not change the distinction between the divergent ordinary sum and a value assigned by regularisation.

Further reading / references

• Smoothed asymptotics: from number theory to QFT, Antonio Padilla and Robert G. C. Smith, the paper discussed in the video.
• What do we get if we sum all the natural numbers?, Tony Padilla’s response to the original Numberphile video.
• This blog probably won’t help, Brady Haran’s collection of explanations and related sources about the divergent sum.