Random multiplicative functions exhibit better than square root cancellation

You need to know: Basic probability theory, independent random variables, selection uniformly at random. Notations: ${\mathbb N}$ for the set of positive integers, ${\mathbb C}$ for the set of complex numbers, $|z|$ for the absolute value of complex number z, $\sum\limits_{n\leq x}$ for the sum over positive integers up to x, ${\mathbb E}$ for the expectation, small o notation.

Background: A function $f:{\mathbb N} \to {\mathbb C}$ is called completely multiplicative, if $f(1)=1$ and $f(x \cdot y) = f(x) \cdot f(y)$ for all positive integers x, y. To define such a function, it suffices to define $f(p)$ for primes p. We say that completely multiplicative $f$ is (Steinhaus) random is values $f(p)$ are selected independently, uniformly at random from the unit circle $\{z \in {\mathbb C}:\,|z|=1\}$ . For $0\leq q \leq 1$ , define $g_q(x)=\left(\frac{x}{1+(1-q)\sqrt{\log\log x}}\right)^q$ .

The Theorem: On 20th March 2017, Adam Harper submitted to arxiv a paper in which he proved the existence of positive constants $c,C$ , and $x_0$ , such that Steinhaus random multiplicative function f satisfies $c g_q(x) \leq {\mathbb E}\left|\sum\limits_{n\leq x}f(n)\right|^{2q} \leq Cg_q(x)$ for all $0\leq q \leq 1$ and all $x\geq x_0$ .

Short context: Many functions of central importance in number theory are multiplicative. In many applications, it is important to estimate moments of such functions, that is, expressions of the form $\left|\sum\limits_{n\leq x}f(n)\right|^{2q}$ . The Theorem estimates (up to a constant factor) the moments of a typical multiplicative function. With $q=1/2$ , it implies that $c\frac{\sqrt{x}}{(\log \log x)^{1/4}} \leq {\mathbb E}\left|\sum\limits_{n\leq x}f(n)\right|\leq C\frac{\sqrt{x}}{(\log \log x)^{1/4}}$ . This confirms a conjecture of Helson, who predicted that ${\mathbb E}\left|\sum\limits_{n\leq x}f(n)\right| = o(\sqrt{x})$ . In such cases we say that we have “better than square root cancellation”.