Correlations of multiplicative functions and applications

Abstract

We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots ,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences:\ First, we characterize all multiplicative functions $f:\mathbb{N}\to \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\mathrm{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$ where $a,b$ belong to some multiplicative subsets of $\mathbb{N}.$ This gives a new "circle method-free" proof of the result of Brüdern.

Cited by in corpus (13)

• The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
• Linear correlations of multiplicative functions
• On binary correlations of multiplicative functions
• A note on multiplicative functions resembling the Möbius function
• A note on multiplicative automatic sequences, II
• Correlations of multiplicative functions in function fields
• On the Orbits of Multiplicative Pairs
• Furstenberg systems of pretentious and MRT multiplicative functions
• Convolution of periodic multiplicative functions and the divisor problem
• Partition regularity of Pythagorean pairs
• Beyond the Erdős discrepancy problem in function fields
• Random Chowla's Conjecture for Rademacher Multiplicative Functions
• On variants of Chowla's conjecture