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I was playing with the Collatz Conjecture today, and empirically found a curious behaviour:

Let $S(i)$ be the function that calculates the number of steps needed for $i$ to reach $1$:

It seems that

$\sum\limits_{i=1}^{x} \cos(S(i)) \sim ab\displaystyle\int_{1}^{x} \cos(b\ln(cu)) \,du \sim ax\cos(b\ln(x)), \,\, x \rightarrow \infty, \,\ x \in \mathbb{N}$

Where $\,\,a \approx 0.0754, \,\, b \approx 10.4404, \,\, c \approx 2.102$

The constants have been found empirically while trying to minimize the error between the functions, and do not necessarily correspond to the correct ones.

Graphical evidence up to $2 \cdot10^6$:

Partial sum

Were these facts already known? If not, I conjecture they're true, it's just a matter of finding the correct constants $a,b,c$. Why do you think they behave like this?

Frax
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  • Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/132086/discussion-on-question-by-frax-is-it-already-known-that-sum-i1x-cossi). – Xander Henderson Dec 05 '21 at 23:33
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    Very interesting, although the Collatz Conjecture is unsolved, that is, it is not known that $S(i)$ exists for every $i$. – DanielWainfleet Dec 06 '21 at 21:29

3 Answers3

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Some commenters were concerned with the lack of intermediate points. Here's a plot with all values up to 10000. I think there's definitely something going on here.

plot of sum_i cos collatz steps for i

Interestingly, $\cos(S(i))$ itself has a strong cosine shape.

plot of cos collatz steps for i

Code:

import numpy as np
import matplotlib.pyplot as plt

N = 10000

mem_S = {1:0}

def S(n):
    if n in mem_S:
        return mem_S[n]
    if n % 2 == 0:
        ans = S(n//2) + 1
    else:
        ans = S(3*n + 1) + 1
    return ans

n = np.arange(1, N+1)
S = np.array([S(n) for n in range(1, N+1)])
cos_S = np.cos(S)
sum_cos_S = np.cumsum(cos_S)

plt.plot(n, sum_cos_S)
plt.show()

plt.plot(n, cos_S)
plt.show()

Edit: I have a much better idea of what's going on now. Long story short, this pattern happens because of a numerical coincidence:

$$ 2\pi\frac{\log 2 - \frac{1}{3}\log 6}{\frac{1}{3}\log 6} = 1.0088... \approx 1 $$

For plot 2 above, we can see that $\cos(S(n))$ seems to look "on average" like $\cos(b\log n)$. So the nice looking wave we get for $\Sigma_m^n \cos(S(m))$ just comes from the sum averaging out the noise in $\cos(S(n))$.

The question is why $\cos(S(n))$ seems to look so much like $\cos (b \log n)$ for some $b$.

Here's the plot that answers that question. ($n$ goes up to 100000 in this plot):

scatter plot of S(n) vs log n coloured by e^(iS(n))

The individual points are coloured according to $e^{iS(n)}$. (Command to plot is plt.scatter(np.log(n), S, c=(S % (2*np.pi)), cmap="twilight"))

The first thing to notice is that the points clump up into clusters, and those clusters are organized into a lattice. I have a rough explanation for the existence of the lattice, although the fact that it's so neat and orderly still surprises me. The rough explanation is that for large numbers, the $+1$ in $3n+1$ is tiny compared to the overall size of the number. So generally, we can expect that a $3n+1$ step is more or less like adding $\log 3$ onto $\log n$ and an $n/2$ step is like subtracting $\log 2$. Getting from $n$ to 1 on a linear scale means getting from $\log n$ to 0 on a log scale. So the points of the lattice are given by $(a \log 2 - b \log 3, a+b)$ for integers $a,b$, and the small $+1$ errors create a spread out cluster of points around each lattice point.

Now that the lattice is explained, the next question is why there are vertical stripes of colour, corresponding to $e^{iS(n)}$. It was perfectly possible that these stripes not be vertical, but some other angle. That would cause the pattern to be destroyed by destructive interference.

The lattice basis vectors given by the expression above are $(\log 2, 1)$ and $(-\log 3, 1)$. We can change basis to replace $(-\log 3, 1)$ with $(\log 2 + \log 3, 0) = (\log 6, 0)$. The $S(n)$ component of this vector is 0, so lattice points separated by this vector all have the same value of $S(n)$, and thus the same value of $e^{iS(n)}$.

So the pattern must have $\log 6$ periodicity along the $\log n$ axis. But in fact, the wavelength we observe is $\frac{1}{3} \log 6$, a special case of $\log 6$ periodicity.

We can now use the remaining basis vector, $(\log 2, 1)$ to check if the stripes are vertical. Incrementing $S(n)$ changes the phase $e^{iS(n)}$ by a factor of $e^i$. If the stripes are vertical, then adding $\log 2$ to $\log n$ should change the phase by $e^{2\pi i \log 2 / \frac{1}{3}\log 6}$. Since,

$$ 2\pi\frac{\log 2 - \frac{1}{3}\log 6}{\frac{1}{3}\log 6} = 1.0088... \approx 1 $$

these phase differences are approximately equal, as expected. This implies the stripes are indeed approximately vertical. We could get exactly vertical stripes by scaling $S(n)$ very slightly before applying the cosine. Without this correction, the formula will eventually fail for large $n$. This method also gives a value for $b$, being $2\pi$ over the wavelength:

$$ b = \frac{6\pi}{\log 6} = 10.5201... $$

Ricky Tensor
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    The first plot is of the partial sums? It does look remarkably smooth zoomed out. | Betraying my apparently embarrassing ignorance of the term "scatter plot": between the 2nd and 3rd plots, which one is $y=\cos S(x)$ and what is the other representing? – runway44 Dec 05 '21 at 06:36
  • @runway44 That's correct about plot 1. Plots 2 and 3 both represent $cos(S(x))$, that is, they both have the same data points. It's just that plot 2 has lines drawn between adjacent data points, and plot 3 doesn't. – Ricky Tensor Dec 05 '21 at 08:01
  • Also might be interesting to look at $\cos \omega S(x)$ and $\sin\omega S(x)$ for different frequencies $\omega$. – runway44 Dec 06 '21 at 01:42
  • @RickyTensor Look at this Desmos: https://www.desmos.com/calculator/kyrqafmk22?lang=it I added your value, black dots are the first local maxima of the Sum, it seems very unlikely that your $b$ is correct, even though your analysis is truly interesting, uhm... – Frax Dec 06 '21 at 09:35
  • @RickyTensor You may be interested in my recent finding, I found a way to convert $S(i)$ in terms of $b\log(di)$, where $d$ is a new heuristically calculated constant. – Frax Dec 06 '21 at 16:43
  • @Frax Right, I think that value of $b$ only works for the adjusted case, $\sum_m \cos(S(m) \times 1.008818)$. It's only approximate if the stripes aren't vertical. – Ricky Tensor Dec 06 '21 at 20:15
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Here is non-rigorous heuristic approach-

The values of $S(x)$ are random locally, but we can make a global estimate $s(x)$. In half the cases, $x$ is even, where we get $S(x)=1+S(\frac{x}{2})$. In the other half of cases, we get $S(x)=2+S(\frac{3x+1}{2})$. Thus: $$s(x)=\frac{1}{2}(1+s(\frac{x}{2})) + \frac{1}{2}(2+S(\frac{3x}{2}))$$ Taking $u=\frac{3x}{2}$, we have: $$s(u) = 2s(\frac{2u}{3}) - s(\frac{u}{3}) - 3$$ We can see that $s(x)$ must grow logarithmically. Let us estimate $s(x)=b\log{x}$. Then, our equation above gives us the constraint: $$b\log{u} = 2b(\log{u}+\log{2}-\log{3}) - b(\log{u}-\log{3}) - 3$$ $$b(2\log{2}-\log{3}) = 3 \implies b = \frac{3}{2\log{2}-\log{3}} \approx 10.43$$

Now, we have a rough estimate for $S(i)$, given by $s(i)$. Let's try to use this to estimate our summation: $$f(x) = \sum_{i=1}^x \cos(S(i)) \approx \sum_{i=1}^x \cos(s(i)) = \sum_{i=1}^x \cos(b\log{i}) \sim \int_1^x \cos(b\log{u}) du$$

Evaluating the integral, we get: $$f(x) \approx \frac{bx\sin(b\log{x})+\cos(b\log{x})}{b^2+1} \sim \bigg(\frac{b}{b^2+1}\bigg)x\sin(b\log{x})$$

Since phase changes do not affect the estimation much, we get something of the form $ax\cos(b\log{x})$, where $a=\frac{b}{b^2+1} \approx 0.095$.

Our values of $b$ match, but the values of $a$ do not. I am unsure why this happens. Maybe this method can be refined to get more accurate values, or I have made a mistake in the calculation, or the experimental value of $a$ will approach the estimated value.

Haran
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  • Interesting, I'm testing both your values in my program – Frax Dec 05 '21 at 11:45
  • Yes, your $b$ is working in a sense, by your $a$ is too big – Frax Dec 05 '21 at 11:48
  • I am not sure why. Maybe someone else could read my answer and provide some suggestions on how to improve it and some insight on where the mistake could be. – Haran Dec 05 '21 at 11:49
  • I'm not sure, but your $b$ might be too small, the error as $x \rightarrow \infty$ "seems" to increase, but maybe I'm wrong. – Frax Dec 05 '21 at 12:01
  • I computed the graphic up to $10^8$ and modified $a$ and $b$ accordingly to reduce the new error. – Frax Dec 05 '21 at 15:07
  • It occurs to me though that it is easily possible for $a(i)\sim b(i)$ while $\sum \cos a(i)$ and $\sum \cos b(i)$ are as far apart as they could possibly be. – runway44 Dec 05 '21 at 22:56
  • @runway44 not when $a(i)$ and $b(i)$ grow sublinearly though – Bananach Dec 05 '21 at 23:12
  • Btw, according to an article I linked in my answer, the "correct" value of $b$ is $6.95$, although that's for a typical value, not including outliers. I can't find any glaring first order mistakes with your derivation myself though – Bananach Dec 05 '21 at 23:13
  • @Bananach If $a(i)$ and $b(i)$ are not at most $o(1)$ apart (which necessarily applies here, because $S(i)$ is always an integer and $\Delta s(i)$ is $o(1)$), then the difference between $\sum \cos a(i)$ and $\sum \cos b(i)$ could be up to $\Omega(n)$ apart, meaning the error could be in the same order of magnitude as the estimate itself. – runway44 Dec 05 '21 at 23:17
  • There is no way $6.95$ is the answer for $b$, as of $10^8$ I'm "empirically" sure of my $b$ down to around $10.43\pm 0.02$ – Frax Dec 05 '21 at 23:32
  • Btw, even though your $a$ seems very different from my value, it's not impossible that with $x \rightarrow \infty$ the sum will "expand" enough to accomodate it, that would also prove your $b$. Tomorrow I'll make my program test indefinitely long your two constants to see how the error goes as $x \rightarrow \infty$ – Frax Dec 06 '21 at 00:00
  • @runway44 you're right – Bananach Dec 06 '21 at 08:46
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Differentiating both $\sum S(i)$ and $i\cos\log i$ and then taking arc cosines roughly translates your claim to $S(i)\sim \log(i)$, which is plausible if you believe multiplications by three and divisions by two appear at a roughly equal ratio in the Collatz sequence for all starting values.

In fact, we clearly have $S(i)\geq \log_2(i)$ and your conjecture may be equivalent to a strong Collatz conjecture of the form that $S(i)$ is not only finite but behaves like $S(i)\leq C\log i$ in some asymptotic and amortized sense.

PS: undoing your operations to handwaveily reduce your conjecture to something more intuitively believable is in no way supposed to take away from your discovery. In fact, maths is often about transforming a problem until it suddenly looks manageable, and the comparison of the scatter plots of $S(i)$ and of $\cos(S(i))$ in the other answer shows that your transformations do appear to reveal a cleaner, more manageable, structure. Indeed, the fact that the values of $\cos(S(i))$ change phase continuously, even though the $S(i)$ don't, is very interesting to me and I'm hoping someone that knows number theory better than I can elucidate.

Edit: according to https://www.ams.org/journals/mcom/2003-72-242/S0025-5718-02-01425-4/home.html researchers conjecture that $S(i)\sim 6.95log(i)$ for most values of $S(i)$.

Bananach
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  • Interesting analysis, I obviously imagined it could be reduced, it would be very presumptuous to think otherwise! My current attention is on $a$ and $b$, I wonder if they can be found, not only empirically. I wonder if they are even rational, but I don't think so. – Frax Dec 05 '21 at 10:49
  • Check the article I added to my answer – Bananach Dec 05 '21 at 23:10
  • that is a very different value from my $b$, I wonder why, maybe $a$ weights in a particular way such that inside the cosine there must be a bigger constant. – Frax Dec 05 '21 at 23:24
  • The Wikipedia article on the Collatz conjecture provides a plot of $S(i)$ which shows the graph is fractal-like, so I don't think an estimate like this one makes sense. Even if it did, it is possible to vary a sequence by $O(1)$ and change the value of the partial sums of its cosines by $\Omega(n)$, so it's not clear to me how the estimate for $S(i)$ could have been used. – runway44 Dec 06 '21 at 01:38