VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES

Abstract

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set$A$whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern$n+1\in A$,$n+2\in A$,$n+3\in A$is positive as long as$A$has density greater than$\frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of$A$having density exactly$\frac{1}{3}$, below which one would need nontrivial information on the local distribution of$A$in Bohr sets to proceed. We apply our results first to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors$P^{+}(n)$,$P^{+}(n+1),P^{+}(n+2)$of three consecutive integers. Second, we show that the tuple$(\unicode[STIX]{x1D714}(n+1),\unicode[STIX]{x1D714}(n+2),\unicode[STIX]{x1D714}(n+3))~(\text{mod}~3)$takes all the$27$possible patterns in$(\mathbb{Z}/3\mathbb{Z})^{3}$with positive lower density, with$\unicode[STIX]{x1D714}(n)$being the number of distinct prime divisors. We also prove a theorem concerning longer patterns$n+i\in A_{i}$,$i=1,\ldots ,k$in approximately multiplicative sets$A_{i}$having large enough densities, generalizing some results of Hildebrand on his ‘stable sets conjecture’. Finally, we consider the sign patterns of the Liouville function$\unicode[STIX]{x1D706}$and show that there are at least$24$patterns of length$5$that occur with positive upper density. In all the proofs, we make extensive use of recent ideas concerning correlations of multiplicative functions.