Multiplicative functions in short intervals, with applications
In the paper under review the author explains recently developed methods of studying and understanding the behaviour of multiplicative functions in short intervals, due to herself and M. Radziwiłł, T. Tao, J. Teräväinen, and T. Ziegler. Most of results can be understood, as well, in terms of the Liouville function \(\lambda:\mathbb{N}\mapsto\{-1,1\}\), defined by \(\lambda(n)=(-1)^k\) where \(k\) is the number of prime factors of \(n\) (counted with multiplicity). The key and motivating result about this function, which is actually equivalent to the prime number theorem, asserts that ``about half of the numbers have an odd number of prime factors and half of the numbers have an even number of prime factors''. More precisely, \(\sum_{n\leq X}\lambda(n)=o(X)\) as \(X\to\infty\). The central question of the paper is: what will happen if we restrict the above sum to run over the so-called short interval \((x,x+H]\)? After a challenging time-line of improving results in this direction, as reviewed in the paper, [the author and \textit{M. Radziwiłł}, Ann. Math. (2) 183, No. 3, 1015--1056 (2016; Zbl 1339.11084)] made a breakthrough on understanding the Liouville function in short intervals by proving that if \(X\geq H\geq 2\) and \(H\to\infty\) with \(X\to\infty\), then \[ \sum_{x<n\leq x+H}\lambda(n)=o(H) \] for almost all \(x\leq X\). This unconditionally improves upon a known result that was conditional on the Riemann hypothesis. The method of proving this results does work for more general multiplicative functions \(f:\mathbb{N}\mapsto [-1,1]\), for which they prove that \[ \left|\frac{1}{H}\sum_{x<n\leq x+H}f(n)-\frac{1}{X}\sum_{n\leq X}f(n)\right|=o(1) \] for almost all \(x\leq X\). An analogue result for complex-valued functions also obtained very recently. The author reports several applications, including studying intervals containing smooth numbers, finding a lower bound for the number of sign changes of the Liouville function, and more precisely Chowla's conjecture concerning correlations of the Liouville function, and generalized correlations of multiplicative functions.
average of multiplicative function on short interval
scientific article; zbMATH DE number 7314414
scientific article