Rigidity theorems for multiplicative functions

Abstract

We establish several results concerning the expected general phenomenon

On completely multiplicative automatic sequences

“…As an application, all multiplicative and automatic sequences produced by primitive automata are Weyl rationally almost periodic. We remark that a sequence (b n ) n∈N is called Weyl rationally almost periodic if it can be approximated by periodic sequences over same alphabet in the pseudo-metric In [KM17], the authors considered general multiplicative functions with the condition lim inf N →∞ |b n+1 − b n | > 0. They proved that if (b n ) n∈N is a completely multiplicative sequence, then most primes, at a fixed power, give the same values as a Dirichlet character.…”

The twin prime conjecture

“…Although new ideas appear to be required to prove the Twin Prime Conjecture, the breakthroughs of Theorems 7, 8 and 9 have already had several further applications, including new results on large gaps between primes [21,20,52], the resolution of the Erdős discrepancy problem [65], as well as many other results on the distribution of primes [67,51,69,12,6,28,57,42,2,49,3,16,72,70,55,56,33,4,5,1,37,53,58,1,45,43] and correlations of multiplicative functions [68,48,39,38,30,41,27,26].…”

On the orbits of multiplicative pairs

“…(ii) if f and g are both pseudo-pretentious in the above sense, say f is pretentious to h 1 n it and g is pretentious to h 2 n it ′ then it may be that t = t ′ , and we must then deal with the distribution in argument of the twist n i(t−t ′ ) , unlike in the outline of the proof of Theorem 1.4 of [9]. With some additional ideas, we are able to address these issues.…”