Rigidity theorems for multiplicative functions

Abstract

We establish several results concerning the expected general phenomenon that, given a multiplicative function $$f:\mathbb {N}\rightarrow \mathbb {C}$$ , 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 $$f:\mathbb {N}\rightarrow \mathbb {T}$$ we have $$\begin{aligned} \liminf _{n\rightarrow \infty }|f(n+1)-f(n)|=0. \end{aligned}$$ Second, we settle an old conjecture due to Chudakov (On the generalized characters. In: Actes du Congres International des Mathematiciens (Nice, 1970), Tome 1, p. 487. Gauthier-Villars, Paris) that states that any completely multiplicative function $$f:\mathbb {N}\rightarrow \mathbb {C}$$ 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 Erdős 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 pretentious multiplicative functions developed earlier by the first author and Szemeredi’s theorem for long arithmetic progressions.