Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Moments of random multiplicative functions over function fields

Maximilian C. E. Hofmann, Annemily Hoganson, Siddarth Menon, William Verreault, Asif Zaman

Abstract

Granville-Soundararajan, Harper-Nikeghbali-Radziwill, and Heap-Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment.

Moments of random multiplicative functions over function fields

Abstract

Granville-Soundararajan, Harper-Nikeghbali-Radziwill, and Heap-Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment.
Paper Structure (16 sections, 21 theorems, 192 equations)

This paper contains 16 sections, 21 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Organization
  5. Preliminaries
  6. Notation
  7. Setup
  8. Remarks on the integer setting
  9. Fourth moment computation using a combinatorial approach
  10. Combinatorial method in the large $q$ limit: proof of \ref{['thm:q-limit']}
  11. Fourth moment using an analytic approach
  12. Analytic Computation of Steinhaus Moments: proof of \ref{['thm:steinhaus']}
  13. Proof of \ref{['prop:convergence_of_B']}: Convergence of $B$ series
  14. Proof of \ref{['prop:integral_representation_of_2kth_moment']}: Integral representation
  15. Proof of \ref{['prop:magic_square_integral']}: magic squares
  16. ...and 1 more sections

Key Result

Theorem 1.1

Fix $k\in \mathbb{N}$. For any prime power $q \geq 2$ and any integer $N \geq 1$, if $f$ is a Steinhaus random multiplicative function defined over $\mathbb{F}_q[t]$, then where $\mathcal{S}_k(N)$ is the number of $k \times k$ magic squares with magic constant $N$, and

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Theorem 1.3
  • Lemma 3.1: Proposition 3 in CORTEEL1998186
  • proof : Proof of \ref{['thm:k=2']}
  • Lemma 4.1: Corollary 3 in LiebProbabilityEstimates
  • Lemma 5.1
  • proof
  • ...and 33 more