Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On variants of Chowla's conjecture

Krishnarjun Krishnamoorthy

TL;DR

This work investigates d-point correlations of completely multiplicative {±1}-valued functions by studying shifted convolution sums Λ_P^H(n) and their natural densities κ_P^H. The authors prove a finite-P multiplicative formula κ_P^H = ∏_{p∈P}(1−2η_p^H) with η_p^H = d/(p+1) for non-exceptional primes, and extend this to small infinite P via a limiting argument, along with a rigidity result that prevents nontrivial P from yielding |κ_P^H|=1. A second main theorem provides a strong constraint on possible values of κ_P^H using combinatorial and two-point correlation arguments, and the spectrum Γ_H of densities is characterized through α_H = inf_p (1−2η_p^H). Groundwork and lemmas on the structure of the N_P^H sets and the algebra A_P underpin these results, while connections to Matomäki–Radziwiłł’s two-point results and Wintner–Wirsing-type density formulas situate the work within the broader landscape of Chowla-type conjectures. Overall, the paper advances an elementary combinatorial approach to Chowla-type questions and clarifies the spectrum of admissible densities arising from shifted convolutions of ±1-valued completely multiplicative functions.

Abstract

We study the shifted convolution sums associated to completely multiplicative functions taking values in $\{\pm 1\}$ and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the corresponding "spectrum".

On variants of Chowla's conjecture

TL;DR

This work investigates d-point correlations of completely multiplicative {±1}-valued functions by studying shifted convolution sums Λ_P^H(n) and their natural densities κ_P^H. The authors prove a finite-P multiplicative formula κ_P^H = ∏_{p∈P}(1−2η_p^H) with η_p^H = d/(p+1) for non-exceptional primes, and extend this to small infinite P via a limiting argument, along with a rigidity result that prevents nontrivial P from yielding |κ_P^H|=1. A second main theorem provides a strong constraint on possible values of κ_P^H using combinatorial and two-point correlation arguments, and the spectrum Γ_H of densities is characterized through α_H = inf_p (1−2η_p^H). Groundwork and lemmas on the structure of the N_P^H sets and the algebra A_P underpin these results, while connections to Matomäki–Radziwiłł’s two-point results and Wintner–Wirsing-type density formulas situate the work within the broader landscape of Chowla-type conjectures. Overall, the paper advances an elementary combinatorial approach to Chowla-type questions and clarifies the spectrum of admissible densities arising from shifted convolutions of ±1-valued completely multiplicative functions.

Abstract

We study the shifted convolution sums associated to completely multiplicative functions taking values in and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the corresponding "spectrum".
Paper Structure (11 sections, 13 theorems, 50 equations)

This paper contains 11 sections, 13 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Completely multiplicative functions
  3. Main results
  4. Brief overview of the proof
  5. Known results
  6. Groundwork
  7. Proof of Theorem \ref{['Theorem "Main Theorem"']}
  8. Proof of Theorem \ref{['Theorem "Main Theorem"']} when $P$ is finite:
  9. Proof of Theorem \ref{['Theorem "Main Theorem"']} when $P$ is a small set:
  10. Proof of Theorem \ref{['Theorem "Second Main Theorem"']}
  11. The spectrum of $\kappa_P^H$

Key Result

Theorem 1

Let $H = \{h_1, \ldots, h_d\}$ be a fixed subset of $\mathbb{N}$. Let $P$ be a small set of primes. Then, for every prime $p$, there exists constants $\eta_p^H$ (depending on $H$) such that Moreover, for every non-exceptional prime, $\eta_p^H = \frac{d}{p+1}$.

Theorems & Definitions (28)

  • Conjecture 1: Chowla
  • Theorem 1
  • Theorem 2
  • Theorem 3: Wintner-Wirsing
  • Theorem 4: Matomäki - Radziwiłł
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 18 more