Rigidity Theorems for Multiplicative Functions

TL;DR

This paper proves new rigidity theorems for multiplicative functions, classifies bounded functions with large gaps, and confirms conjectures relating to their structure and correlations, advancing understanding of their independence and special forms.

Contribution

It classifies bounded completely multiplicative functions with large gaps, proves a conjecture on functions with finite values and bounded discrepancy, and characterizes functions with similar binary correlations to Dirichlet characters.

Findings

Bounded completely multiplicative functions with large gaps are classified.

Finiteness and bounded discrepancy imply the function is a Dirichlet character.

Functions with similar binary correlations to Dirichlet characters are of the form χ'(n)n^{it}.

Abstract

We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form. First, we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values. This implies the solution of the following folklore conjecture: for any completely multiplicative function $f:\mathbb{N}\to\mathbb{T}$ we have $$\liminf_{n\to\infty}|f(n+1)-f(n)|=0.$$ Second, we settle an old conjecture due to N.G. Chudakov [Actes du ICM ({N}ice, 1970), {T}. 1, p. 487] that states that any completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ that: a) takes only finitely many values, b) vanishes at only finitely many primes, and c) has bounded discrepancy, is a Dirichlet character. This generalizes previous work of Tao on the Erd\H{o}s Discrepancy Problem. Finally, we show that if many of the binary correlations of a 1-bounded multiplicative function are asymptotically equal to those of a Dirichlet character $\chi$ mod $q$ then $f(n) = \chi'(n)n^{it}$ for all $n$, where $\chi'$ is a Dirichlet character modulo $q$ and $t \in \mathbb{R}$. This establishes a variant of a conjecture of H. Cohn for multiplicative arithmetic functions. The main ingredients include the work of Tao on logarithmic Elliott conjecture, correlation formulas for \emph{pretentious} multiplicative functions developed earlier by the first author and Szemeredi's theorem for long arithmetic progressions.