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The Multivariate Rate of Convergence for Selberg's Central Limit Theorem

Asher Roberts

Abstract

In this paper we quantify the rate of convergence in Selberg's central limit theorem for $\log|ζ(1/2+it)|$ based on the method of proof given by Radziwill and Soundararajan. We achieve the same rate of convergence of $(\log\log\log T)^2/\sqrt{\log\log T}$ as Selberg in the Kolmogorov distance by using the Dudley distance instead. We also prove the theorem for the multivariate case given by Bourgade with the same rate of convergence as in the single variable case.

The Multivariate Rate of Convergence for Selberg's Central Limit Theorem

Abstract

In this paper we quantify the rate of convergence in Selberg's central limit theorem for based on the method of proof given by Radziwill and Soundararajan. We achieve the same rate of convergence of as Selberg in the Kolmogorov distance by using the Dudley distance instead. We also prove the theorem for the multivariate case given by Bourgade with the same rate of convergence as in the single variable case.
Paper Structure (14 sections, 18 theorems, 125 equations)

This paper contains 14 sections, 18 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Relation to Previous Results
  4. Structure of the Proofs
  5. Proofs
  6. Proof of \ref{['thm:selbergcltrate']}
  7. Proof of \ref{['cor:multivariateselbergclt']}
  8. Proof of \ref{['prop:movingoffaxis']}
  9. Proof of \ref{['prop:mollifying']}
  10. Proof of \ref{['prop:approximatingthemollifier']}
  11. Proof of \ref{['prop:discardingprimes']}
  12. Proof of \ref{['prop:truncatingprimesum']}
  13. Proof of \ref{['prop:comparingmoments']}
  14. Proof of \ref{['prop:comparingnormals']}

Key Result

Theorem 1

Let $\tau$ be a random point distributed uniformly on $[T,2T]$, and let $|h-h'|\sim(\log T)^{-\alpha}$ for $\alpha\in(0,1)$. Then for $T$ large enough, where $(\mathcal{Z},\mathcal{Z}')$ is a Gaussian vector with mean $0$ and covariance matrix $$.

Theorems & Definitions (34)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Proposition 1: Moving off axis
  • Proposition 2: Mollifying
  • Proposition 3: Approximating the mollifier
  • Proposition 4: Discarding higher order primes
  • Proposition 5: Truncating the prime sum
  • Proposition 6: Comparing moments
  • Proposition 7: Comparing normals
  • ...and 24 more