A number called 4/3
Just looking at the characteristic function
of the Syracuse variable
A random variable sits at the heart of the Collatz conjecture. Where does the peak of its characteristic function stand? A record of pushing n up to 21 by computer and simply watching. There is no proof here. Only an observation.
This is a note, not a paper.
Between work, I play with number theory as a hobby. I like problems that wear a simple face and yet no one can crack — primes, Collatz — and on my days off I let the computer run, a little at a time. This was that kind of thing: weekend carpentry, you could say. I just wanted to see, with my own eyes, the characteristic function of the random variable at the heart of Tao's proof, so I started raising n from 1 — and before I knew it I was at n=21, a computation over seven billion points.
What I saw was cleaner than I expected. The position of the peak walks straight ahead, at slope 4/3. Since I'd come this far, I tidied up the observation and put it, code and numerical data and all, on Zenodo with a DOI (link below) — so that anyone who wants to carry the computation further can reproduce it as is.
I have proven nothing. This contributes nothing to the Collatz conjecture. I just found what I glimpsed inside interesting enough to write down.
First, the 3N+1 story
Pick a positive integer. If it's even, halve it. If it's odd, triple it and add one. Repeat — and no matter where you start, you eventually reach 1. Or do you?
This is the Collatz conjecture. Since 1937, no one has proven it. The question could hardly be simpler, and yet it has defeated everyone.
In 2019, Terence Tao blew a large hole in the problem. He showed that "almost all" orbits reach almost bounded values. At the technical heart of that proof sits an object called the Syracuse random variable.
What does "well spread out" mean? In Fourier's language: the peak of the characteristic function (the Fourier transform) is low. If the peak is high, probability mass is concentrated at that frequency. If it's low, the mass is spread flat. Tao proved that this peak sinks with n "faster than any polynomial."
But it doesn't say where the peak stands.
2 is the vowel of the world of 3
To talk about the "position" of the peak of the characteristic function, you need a coordinate. Here a classical fact of elementary number theory comes into play.
For every n, 2 is a primitive root modulo 3n. That is, every frequency ξ not divisible by 3 can be written in exactly one way as a power of 2, ξ = 2k mod 3n. Keep cycling the powers of 2 and you step through every frequency coprime to 3, each exactly once, leaving none out.
So I stop viewing frequencies by ξ and instead view them by the exponent k. This is the whole viewpoint of the note. I reindex the characteristic function as a function of k, and call it ψn(k). Then I write the k at which its absolute value is largest as k*(n).
The peak walks at slope 4/3
Raising n from 1 to 21, I computed ψn over its whole domain. At n=21 the domain reaches 2·320 ≈ 7 billion points. Record the peak position k*(n) at each n and line them up, and — neatly, almost in a straight line — they climb. The slope is right around 4/3.
24 ÷ 18 = 4/3. With zero error.
A formula right 14 times in a row breaks at 18
Drop the slope 4/3 into an integer prediction and you get a simple floor function, ⌊4(n−1)/3⌋. Line it up and at first it's almost eerily accurate. From n=4 to 17, 14 exact matches in a row. One look and you want to believe it's a structural identity.
But at n=18, for the first time, it missed. Against a prediction of 22, the measured value was 23. A gap of just 1. Then +1 at n=20, and +2 at n=21. The set of n where it misses is {2, 3, 18, 20, 21} — clustered at the ends of the range.
What's interesting is that even when the formula misses, the slope 4/3 doesn't break. The gap at n=21 is +2, yet k*(21)/21 = 4/3 still holds exactly. A +1 "wobble" shows up only at the boundary of the floor function; the underlying linearity is preserved. The formula is only an approximation, but the slope looks real.
How the height of the peak sinks
Not just the position — I looked at the height too. Plot the peak value Mn = max|ψn| against n and it decays smoothly. Fit it to a line on a log-log plot, and over the finite range (n=9–21) a power law Mn ∼ C·n−2.1 fits well.
But this is a finite-range observation, not an asymptotic claim. Tao's upper bound guarantees superpolynomial decay — faster than any fixed power n−A. The "effective exponent 2.1" seen through a finite window is consistent with that exponent slowly growing as n increases. At the present range, you can't see that far.
The numbers themselves
I turned it into figures, but here's the raw table too. The whole of the observation is in it. gn/Mn2 is a measure of how far the peak pulls ahead of the runner-up.
| n | Mn | k*(n) | ⌊4(n−1)/3⌋ | gap | gn/Mn2 |
|---|---|---|---|---|---|
| 1 | 5.7735e−1 | 0 | 0 | · | — |
| 2 | 3.7792e−1 | 2 | 1 | +1 | 0.654 |
| 3 | 2.5224e−1 | 3 | 2 | +1 | 0.406 |
| 4 | 1.7700e−1 | 4 | 4 | · | 0.421 |
| 5 | 1.2927e−1 | 5 | 5 | · | 0.394 |
| 6 | 9.6106e−2 | 6 | 6 | · | 0.106 |
| 7 | 7.5870e−2 | 8 | 8 | · | 0.105 |
| 8 | 6.0891e−2 | 9 | 9 | · | 0.220 |
| 9 | 4.8026e−2 | 10 | 10 | · | 0.119 |
| 10 | 3.8278e−2 | 12 | 12 | · | 0.027 |
| 11 | 3.1944e−2 | 13 | 13 | · | 0.133 |
| 12 | 2.6458e−2 | 14 | 14 | · | 0.111 |
| 13 | 2.2052e−2 | 16 | 16 | · | 0.025 |
| 14 | 1.9128e−2 | 17 | 17 | · | 0.139 |
| 15 | 1.6284e−2 | 18 | 18 | · | 0.014 |
| 16 | 1.4409e−2 | 20 | 20 | · | 0.109 |
| 17 | 1.2511e−2 | 21 | 21 | · | 0.023 |
| 18 | 1.1187e−2 | 23 | 22 | +1 | 0.103 |
| 19 | 9.8157e−3 | 24 | 24 | · | 0.009 |
| 20 | 8.8846e−3 | 26 | 25 | +1 | 0.110 |
| 21 | 7.8721e−3 | 28 | 26 | +2 | 0.012 |
All in IEEE 754 double-precision complex arithmetic. Cross-checked against an independent implementation (recursion in the original space → discrete Fourier transform) for n≤8, agreeing to a relative error below 10−10. n=20, 21 were computed on a cloud machine with 961 GB of memory.
What I don't know
This note has no answers. It sets down an observation and leaves the questions open. That, exactly, is a numerical open question.
I leave this observation as a record, not a conclusion. To the Collatz conjecture itself, it contributes nothing. I just wanted, once, to look properly at how the object at the heart of Tao's proof behaves inside a computer. Push n up to 21 and the peak walks at slope 4/3; the floor function staggers by 1 somewhere along the way, then comes back to 4/3. That's all of it.
Do this kind of thing all the time and people occasionally ask why. The answer is simple: I want to see, with my own eyes, the regularity hidden behind a simple question. It isn't useful for anything. That seven billion points of computation turned up the plain fraction "4/3" — that single point — is, I think, worth a few weekends. Maybe something will come of it someday; maybe combining things that seem unrelated will lead to a big discovery.
If anyone can run the computation past n=22, I'd love to see how it continues. The code and the data are there for exactly that.