Rigidity Theorems for Multiplicative Functions

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.