Short and long averages of a bounded multiplicative function are effectively related

You need to know: Notation ${\mathbb N}$ for the set of positive integers, coprime integers.

Background: A function $f:{\mathbb N} \to [-1,1]$ is called multiplicative if $f(1)=1$ and $f(ab)=f(a)f(b)$ holds for all coprime $a,b\in {\mathbb N}$ .

The Theorem: On 19th January 2015, Kaisa Matomäki and Maksym Radziwiłł submitted to arxiv a paper in which they proved the existence of absolute constants $B,C$ (one can take $B=20000$ ) such that for any multiplicative function $f:{\mathbb N} \to [-1,1]$ , for any $2\leq h \leq X$ , and any $\delta>0$ , inequality $\left|\frac{1}{h}\sum\limits_{x\leq n \leq x+h}f(n) - \frac{1}{X}\sum\limits_{X\leq n \leq 2X}f(n) \right|\leq \delta + B\frac{\log\log h}{\log h}$ folds for all but at most $CX\left(\frac{(\log h)^{1/3}}{\delta^2h^{\delta/25}} + \frac{1}{\delta^2(\log X)^{1/50}}\right)$ integers $x\in[X,2X]$ .

Short context: Many functions of central importance in number theory (for example, the Möbius function and the Liouville function) are multiplicative. For many applications, it is important to estimate average value $\frac{1}{h}\sum\limits_{x\leq n \leq x+h}f(n)$ of such function in short intervals of length h. The Theorem states that this is approximately equal to the average $\frac{1}{X}\sum\limits_{X\leq n \leq 2X}f(n)$ over “long” interval $[X,2X]$ , which is much easier to estimate. This has a lot of applications, some of them are derived in the same paper.