Sarnak’s conjecture

Prime factors assign each positive integer a Möbius value: zero if a prime factor repeats, otherwise minus one for an odd number of distinct prime factors and plus one for an even number. Set the value at 1 to plus one. Thus the values from 1 through 6 are $1,-1,-1,0,-1,1$: 4 has a repeated factor, while 6 has two distinct factors.

Compare this arithmetic list with observations of a repeated rule, such as the horizontal position of a clock hand advancing a fixed angle each tick. Multiply corresponding entries and average the first N products. Sarnak asks whether that average always approaches zero when the rule has zero topological entropy: its distinguishable patterns do not grow exponentially as we watch longer. The claim covers every continuous observation and every starting state.[1]

Can a low-complexity repeated motion maintain a nonzero average correlation with signs and zeros determined by prime factors? Sarnak's conjecture says no. The average must vanish for every starting point and continuous observation, for every compact topological system with zero entropy.

The arithmetic sequence is the Möbius function. It has $\mu(1)=1$; a repeated prime factor gives zero, and a product of k distinct primes gives $(-1)^k$. For n from 1 through 6, the values are $1,-1,-1,0,-1,1$. Against a constant observation of 1, their first-six average is $-1/6$. This is an example of the calculation, not evidence for its eventual limit.

Let T be the repeated rule on a compact space X, let x be the starting state, and let the continuous function f turn a state into a numerical observation. A circle rotating by a fixed angle, with f recording horizontal position, is a concrete example. Then $f(T^n x)$ is the observation after n steps. The target is $$\lim_{N\to\infty}\frac1N\sum_{n=1}^{N}\mu(n)f(T^n x)=0.$$ The sum uses equal weights for the first N products, an ordinary Cesàro average. Zero topological entropy requires subexponential growth of distinguishable length-N orbit patterns at every fixed observational resolution. The quantifiers cover every continuous f and every x, rather than only typical starting points.[1]

Some systems are covered; the statement for all such systems is open.[1]

March 2026 results cover specified infinite-dimensional torus flows. Restricted systems and other averaging conventions do not settle the universal ordinary-average statement.[1]

This connects patterns of prime factors to the complexity of repeated motion. A stationary system gives a constant observation, so the test becomes the average of the Möbius values themselves. Their long-run cancellation is equivalent to the prime number theorem, the law describing how primes thin out. The full question asks how far that arithmetic cancellation extends when the weights come from a low-complexity motion.[1]

For a stationary system and $f=1$, the displayed average is simply the mean of $\mu(n)$. Its convergence to zero is equivalent to the prime number theorem: the number of primes up to N is asymptotic to $N/\log N$. Although the Möbius values use composite factorizations too, their cancellation is another way to express that prime-distribution law.

The full conjecture tests the same cancellation after weighting by an entire class of deterministic records. Thus it asks whether the growth of possible orbit patterns, rather than the details of one particular motion, separates low-complexity dynamics from persistent correlations with prime-factor arithmetic.[1]