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Rigidity Theorems for Multiplicative Functions

Research paper by Oleksiy Klurman, Alexander P. Mangerel

Indexed on: 25 Jul '17Published on: 25 Jul '17Published in: arXiv - Mathematics - Number Theory



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. Firstly, we prove a converse theorem that resolves the following folklore conjecture: for any completely multiplicative function $f:\mathbb{N}\to\mathbb{T}$ we have $$\liminf_{n\to\infty}|f(n+1)-f(n)|=0.$$ Secondly, we settle an old conjecture due to N.G. Chudakov 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 uniformly bounded partial sums, is a Dirichlet character. 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.