Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Magic squares, the symmetric group and Möbius randomness

Ofir Gorodetsky

Abstract

Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the $k$-fold divisor function.

Magic squares, the symmetric group and Möbius randomness

Abstract

Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the -fold divisor function.

Paper Structure

This paper contains 9 sections, 8 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Proof of the Diaconis--Gamburd Theorem
  3. The symmetric group
  4. The characteristic map
  5. Conclusion of proof
  6. Symmetric powers
  7. General moments
  8. Number-theoretic connections
  9. A polynomial analogue of dlambda

Key Result

Proposition 2.3

Suppose $\mu,\widetilde{\mu}\vdash n$. We have $N_{\mu,\widetilde{\mu}}' =N_{\mu,\widetilde{\mu}}$.

Theorems & Definitions (22)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • ...and 12 more