---
title: 'Researchers Thought This Was a Bug (Borwein Integrals)'
source: 'https://youtube.com/watch?v=851U557j6HE'
video_id: '851U557j6HE'
date: 2026-08-10
duration_sec: 1045
---

# Researchers Thought This Was a Bug (Borwein Integrals)

> Source: [Researchers Thought This Was a Bug (Borwein Integrals)](https://youtube.com/watch?v=851U557j6HE)

## Summary

The video explores the Borwein integrals, a sequence of definite integrals that appear to equal π for a long time but deviate slightly at a specific point. It explains the phenomenon using Fourier transforms and convolutions, revealing a beautiful mathematical structure.

### Key Points

- **The Sequence of Integrals** [00:00] — The sequence of integrals starts with sinc(x) and multiplies by sinc(x/3), sinc(x/5), etc., each equaling π/2.
- **The Break at 15** [03:12] — The pattern breaks at the factor sinc(x/15), where the integral becomes slightly less than π/2.
- **Fourier Transform Connection** [12:18] — The Fourier transform of sinc is the rect function, and the integral of a function equals its Fourier transform evaluated at zero.
- **Convolution Theorem** [15:17] — The convolution theorem states that the Fourier transform of a product is the convolution of the individual transforms, which acts as a moving average.
- **Why 15?** [08:20] — The pattern breaks when the sum of reciprocals of odd numbers exceeds 1, which first happens at 1/15.

## Transcript

I have up on screen here a sequence of computations, and don't worry, in a moment we're gonna unpack and visualize what each one is really saying. What I want you to notice is how the sequence follows a very predictable,
if random, seeming pattern, and how each computation happens to equal pi. you might think that this was a pattern that would go on forever. At some point it stops, and instead of equaling pi,
you get a value which is just barely, barely less than pi. The main character in the story today is the function sine of x divided by x.
This actually comes up commonly enough in math and engineering that it gets its own name, sinc, and the way you might think about it is by starting with a normal far away from zero by multiplying it by 1 over x.
since when you plug that in it looks like dividing 0 by 0. And then the even more astute among you, maybe fresh out of a calculus class, the function gets closer and closer to 1.
you get a nice continuous curve. All of that is a little by the by because the thing we actually care about is the which you'd think of as meaning the area between the curve and the x-axis,
or more precisely the signed area, meaning you add all the area bound by the positive parts of the graph in the x-axis, and you subtract all of the parts bound by the Like we saw at the start, it happens to be the case that this evaluates to be exactly pi,
you would approach this with the usual tools of calculus. Progressing on with the sequence I opened with,
the next step is to take a copy of the sinc function, where you plug in x divided by 3, but stretched out horizontally by a factor of 3. we get a much more complicated wave whose mass seems to be more concentrated
completely changes the area. You can't just randomly modify an integral like this and expect nothing to change. that nothing has changed.
And the next step in the sequence was to take an even more stretched out version of the sinc function by a factor of 5, multiply that by what we already have, and again look at the signed area underneath the whole curve, which again equals pi.
With each iteration, we stretch out by a new odd One thing you might notice is how except at the input x equals 0, every single part of this function is progressively getting multiplied by something
So you would expect, as the sequence progresses, and if anything you would expect the area to be getting smaller. Eventually that is exactly what happens, but what's bizarre is that it
stays so stable for so long, and of course more pertinently, that when it does break at the value 15, it does so by the tiniest tiny amount. maybe because we're doing something with floating-point arithmetic,
if you work this out more precisely, here is the exact value of that last integral, which is a certain fraction of pi where the numerator and the denominator are absurd. They're both around 400 billion billion billion.
Jonathan and David Borwein, which is very fun, and they mentioned how when a fellow researcher was computing these integrals using a computer algebra system, he assumed that this had to be some kind of bug.
And it gets weirder than that actually. If we take all these integrals and include yet another factor, 2 cosine of x, you can't just randomly multiply new things into an integral like this,
and it's not until you get to the number 113 that it breaks. subtle amount that you could imagine. So the natural question is what on earth is going on here?
And luckily there actually is a really satisfying explanation for all this. first looks completely unrelated, but it shows a similar pattern, You get to the number 15, and then it falters by just a tiny amount.
is secretly the same as all our integral expressions, but in disguise. consider a function that I'm going to be calling rect of x,
which is defined to equal 1 if the input is between negative 1 half and 1 half, So the function is this boring step, basically. so I'll call it f1 of x, and each new function in our sequence
is going to be a kind of moving average of the previous function. So for example, the way the second iteration will be defined is to take this when the window is centered at that input x, the value in my new function drawn
below is defined to be equal to the average value of the first function above So for example, when the window is far enough to the left, every value inside it is 0, so the graph on the bottom is showing 0.
the average value is a little more than 0, and you see that in the graph below. And notice that when exactly half the window is over that plateau at 1 and half of it is at 0, the corresponding value in the bottom graph is 1 half, and you get the point.
The important thing I want you to focus on is how when that window is then the average value is also 1, so we get this plateau on our function at the bottom.
think about is the length of the plateau for that second function. If you think about it for a moment, the distance between the left
will be exactly half of the width of the window, so half of 1 third. the edges of the plateaus is half of the window width.
So overall it's 1 minus that window width, which is 1 minus 1 third. The value we're going to be computing, the thing that will look stable for which in both of these iterations is equal to 1 because it's inside that plateau.
but this time with the window whose width is 1 fifth. It's kind of fun to think about why as you slide around this window you get a the significant thing I want you to focus on is how when that window is entirely
inside the plateau of the previous function, then by definition the bottom function This time the length of that plateau on the bottom will be the length of the previous one, 1 minus 1 third, minus the window width, 1 fifth.
The reasoning is the same as before in order to go from the point where the middle of the window is on that top plateau to where the entirety of the window is inside that and once more the value to record is the output of this function when the input is 0,
The next iteration is a moving average with a window width of 1 seventh. Doing one more iteration with 1 over 9, the plateau gets smaller by that amount.
And also notice how just outside of the plateau the function is really really close to 1 because it's always been the result of an average between the plateau at 1 and the neighbors, which themselves are really really close to 1.
where we're sliding a window with width 1 15th across the whole thing. At that point the previous plateau is actually thinner than the window itself. will have to be ever so slightly smaller than 1.
And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, one third plus one fifth plus one seventh, on and on, it's once we get to one fifteenth that that sum grows to be bigger than 1.
And in the context of our shrinking plateaus, having started with a plateau of width 1, The point is with this as a sequence of functions that we've defined by a
seemingly random procedure, if I ask you to compute the values of all of these functions at the input 0, you get a pattern which initially looks stable. iteration it falls short ever so slightly, just barely.
This is analogous, and I claim more than just analogous, to the integrals we saw earlier, where we have a stable value at pi pi pi pi pi until it falls short just barely. And as it happens, this constant from our moving average process that's ever so slightly
smaller than 1 is exactly the factor that sits in front of pi in our series of integrals. they're quantitatively the same as well. And when it comes to the case where we add the 2 cosine of x term inside the integral,
which caused the pattern to last a lot longer before it broke down, but where the function we start with has an even longer plateau, So as you do this repeated moving average process,
it takes a lot longer for them to eat into the whole plateau. More specifically, the relevant computation is to ask how long do you have to add these reciprocals of odd numbers until that sum becomes bigger than 2?
And it turns out that you have to go until you hit the number 113, which will correspond to the fact that the integral pattern there continues until And by the way, I should emphasize that there is nothing special
That just happens to be the sequence of values highlighted by the Borweins in their paper that made the sequence mildly famous in nerd circles. numbers into those sinc functions, and as long as the sum of
But as soon as they become bigger than 1, our expression drops a little below pi. you can hopefully see why.
these two situations have anything to do with each other? From here, the argument does bring in two mildly heavy bits of machinery, And the way I'd like to go about this is to spend the remainder of this video
without necessarily assuming you're familiar with either of those two topics, and then to explain why the details are true in a video that's dedicated to convolutions, since it's incredibly beautiful and it's useful well beyond this specific,
very esoteric question. where we want to show why the signed area underneath its curve is equal to pi,
we'll make a simple substitution where we replace the input x with pi times x, which has the effect of squishing the graph horizontally by a factor of pi, meaning our new goal is to show why this integral on the right is equal to exactly 1.
refer to this function with the pi on the inside, meaning the area under it is equal to 1. thing as showing the integral on the left, it's just a change of variables.
And likewise for all of the other ones in our sequence, go through each of them, replace the x with a pi times x, and from here the claim is that all these but that both of these are two distinct ways of computing exactly the same thing.
or the engineer sinc function with the pi on the inside, is related to the rect function using what's known as a Fourier transform. there are a few other videos on this channel all about it.
The way it's often described is that if you want to break down a function as a continuous integral of a bunch of pure frequencies, constituent parts.
takes in one function and spits out a new function, and you often think of it as kind of rephrasing the information of your from a new perspective.
where you take a Fourier transform looks like our top hat rect function. This is a nice thing about Fourier transforms for functions that are symmetric
about the y-axis, it is its own inverse, and actually the slightly more general version of our sinc function, where you stretch it horizontally by a factor of k, what you get is a stretched and squished version of this rect function.
unless you can actually do something upon making this translation. And the real idea behind why Fourier transforms are such a useful thing for math and then you look at what they correspond to with respect to the transformed
version of that function, those statements and questions often look very very different in this new language, and sometimes it makes the questions a lot easier For example, one very nice little fact, another thing on our list of things to show,
is that if you want to compute the integral of some function from negative infinity it's the same thing as simply evaluating the Fourier transformed version of that This is a fact that will actually just pop right out of the definition,
and it's representative of a more general vibe that every individual some kind of global information about the original function on the left. In our specific case, it means if you believe me that this sync function and the
rect function are related with a Fourier transform like this, it explains the integral, because it's saying all that signed area is the same thing as evaluating rect at zero,
Now, you could complain, surely this just moves the bump under the rug. would be as hard as computing the original integral. computing these Fourier transforms, and moreover, that when you do,
You get a lot of bang for your buck out of doing the computation. Now, the other key fact that will explain the connection we're hunting for is that if and then you take the sum of the Fourier transform of that product,
it will be the same thing as if you individually took the Fourier transforms of your original function and then combined them using a new kind of operation that we'll talk Now, even though there's a lot to be explained with convolutions,
the upshot will be that in our specific case with these rectangular functions, taking a convolution looks just like one of the moving averages that we've been talking about this whole time, combined with our previous fact that integrating in
one context looks like evaluating at zero in another context, if you believe me that multiplying in one context corresponds to this new operation, convolutions, which for our example you should just think of as moving averages,
that will explain why multiplying more and more of these sinc functions together can be thought about in terms of these progressive moving averages and always evaluating at zero, which in turn gives a really lovely intuition for why you would
expect such a stable value before eventually something breaks down as the edges of the plateau inch closer and closer to the center. It's called the convolution theorem, and again,
I recognize that it's maybe a little unsatisfying to end things here by laying down three magical facts and saying everything follows from those, like Fourier transforms can be so useful for tricky problems.
where hard problems can sometimes look easier. learn about these beautiful things like the convolution theorem.
As one more tiny teaser, another fun consequence of this convolution theorem will be that it opens the doors for an algorithm that lets you compute the product of two large numbers very quickly, like way faster than you think should be even possible.
So with that, I'll see you in the next video.