Multiplicative functions

A function $f: \mathbb{N} \to S^1$ is called completely multiplicative if $f(m\cdot n) = f(m) \cdot f(n)$ . An example is the function $\Omega(n)=q_1 + \cdots + q_k$ if $n=p_1^{q_1}\cdots p_k^{q_k}$ . Note that $\lambda(n)=(-1)^{\Omega(n)}$ is the Liouville function.

In link the authors introduced multiplicative dynamical systems. Given a continuous map $T:X\to X$ , it induces an additive dynamical system $(X, T)$ , in the sense that $T^{m+n} = T^m\circ T^n$ . It also induces a multiplicative dynamical system $(X, T^{\Omega})$ , in the sense that $T^{\Omega(m\cdot n)} = T^{\Omega(m)}\circ T^{\Omega(n)}$ .

More generally, a multiplicative dynamical system is a map $S: \mathbb{N} \to C(X, X)$ such that $S(m\cdot n) = S(m) \circ S(n)$ . It is easy to see that a multiplicative dynamical system is determined by the generators $S(p)$ , $p$ prime. Then $(Y, S)$ is said to be finitely generated if $\{S(p)| p \text{ prime}\}$ is a finite set. In the case $S(n)=T^{\Omega(n)}$ , we see that $S(p)= T$ for every prime $p$ . In particular, it is finitely generated.

Motivated by the Mean Value Theorem for prime numbers, the authors formulated the following (much more generalized) version of Sarnak Conjecture: let $(X, T)$ be an additive dynamical system, $(Y, S)$ a multiplicative dynamical system. Suppose both have low complexity, and there is no local obstruction (that is, aperiodic). Then the two systems are disjoint: for any continuous functions $f$ on $X$ and $g$ on $Y$ , and any points $x\in X$ and $g\in Y$ , the sequences $(f(T^n x))$ and $(g(S_ny))$ are asymptotically independent.

In the particular case that $Y=\{0, 1\}$ , $S_n(k)= k+ \Omega(n) \mod 2$ , $g(k)=(-1)^k$ , the above conjecture reduces to that the average of $(-1)^{\Omega(n)}f(T^n x)$ is $0$ .

The reason that they need to put the low complexity assumption on the dynamical systems is that there are counterexamples. For example, let $T$ be the shift map on $X=\{0,1\}^{\mathbb{N}}$ , $f(x)=(-1)^{x_0}$ , and $\hat x\in X$ be the point generated by $\Omega$ . Then $f(T^n \hat x) =(-1)^{\Omega(n)}$ and hence $(-1)^{\Omega(n)}f(T^n \hat x) =1$ for all $n$ . In particular, the two sequences are the opposite of asymptotically independent.

They also mentioned a fundamental conjecture in this direction: the subshift generated by $\Omega$ has zero entropy. This must be true for the above conjecture to make sense. This might not be an easy question since it is really about the properties of all natural numbers.

Let $\displaystyle \zeta(s)= \sum_{n\ge 1}\frac{1}{n^s}, z\in \mathbb{C}$ be the Riemann zeta function. Euler (before Riemann) observed the product formula for $s>1$ : $\displaystyle \zeta(s)= \prod_{p \text{ prime}}\frac{1}{1- p^{-s}}$ . Then the quotient

$\displaystyle \frac{\zeta(2s)}{\zeta(s)} = \prod_{p} \frac{1- p^{-s}}{1- p^{-2s}} =\prod_{p} \frac{1}{1+p^{-s}} =\sum_{n\ge 1}\frac{\lambda(n)}{n^s}$ , the later being the Dirichlet series for the Liouville function.

Pengfei's Notes