Alexander Mangerel: Correlations, sign patterns and rigidity theorems [...] (NTWS 179)
Channel: Number Theory Web Seminar
Date: 2023-09-11

Title: Alexander Mangerel, Correlations, sign patterns and rigidity theorems for multiplicative functions

Abstract:  The Liouville function \lambda(n), defined to be +1 for n having an even number of prime factors (counted with multiplicity) and -1 otherwise, is a multiplicative function with deep connections to the distribution of primes. Inspired by the prime k-tuples conjecture of Hardy and Littlewood, Chowla conjectured that for every k each of the 2^k distinct sign patterns, i.e., tuples in \{-1,+1\}^k are assumed by the tuples (\lambda(n+1),...,\lambda(n+k)), n \in \mathbb{N}, with the same asymptotic frequency.
The underlying phenomenon at hand is that the prime factorisations of n+1,\ldots,n+k are expected to be (in a precise sense) statistically independent as n varies. As conjectured by Elliott, the same equidistribution of sign patterns is expected to hold for other \pm 1-valued multiplicative functions, provided they are ``far from being periodic''. To the best of our knowledge, until recently no explicit constructions of multiplicative functions with this behaviour were known.
In this talk we will discuss precisely what Chowla's and Elliott's conjectures say, survey some of the literature on correlations, and discuss some related problems about sign patterns. Specifically, we will address:
i) the construction of ``Liouville-like'' functions f: \mathbb{N} \rightarrow \{-1,+1\} whose k-tuples (f(n+1),...,f(n+k)) equidistribute in \{-1,+1\}^k, answering a question of de la Rue from 2018, and
ii) in the case k = 4, the classification of all \pm 1-valued completely multiplicative functions f with the (rigid) property that the sequence of tuples (f(n+1),f(n+2),f(n+3),f(n+4)) omits the pattern (+1,+1,+1,+1), solving a 50-year old problem of R.H. Hudson.
Key to these developments is a new result about the vanishing of correlations of ``moderately aperiodic'' multiplicative functions along a dense sequence of scales.
Based on joint work with O. Klurman and J. Teräväinen.

Link to slides: https://drive.google.com/file/d/1kDLAclRsOTAIspoIdkLUz11L2xcstGUW

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, September 7, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 8, 2023 (1am AEST, 3am NZST)