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Partial sums of typical multiplicative functions over short moving intervals

Mayank Pandey, Victor Y. Wang, Max Wenqiang Xu

Abstract

We prove that the $k$-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval $(x, x+H]$ matches the corresponding Gaussian moment, as long as $H\ll x/(\log x)^{2k^2+2+o(1)}$ and $H$ tends to infinity with $x$. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals $(x, x+H]$ with $H\ll X/(\log X)^{W(X)}$ tending to infinity with $X$, where $x$ is uniformly chosen from $\{1,2,\dots, X\}$, and $W(X)$ tends to infinity with $X$ arbitrarily slowly. This makes some initial progress on a recent question of Harper.

Partial sums of typical multiplicative functions over short moving intervals

Abstract

We prove that the -th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval matches the corresponding Gaussian moment, as long as and tends to infinity with . We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals with tending to infinity with , where is uniformly chosen from , and tends to infinity with arbitrarily slowly. This makes some initial progress on a recent question of Harper.
Paper Structure (9 sections, 15 theorems, 93 equations)

This paper contains 9 sections, 15 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation
  4. Acknowledgements
  5. Moments of random multiplicative functions in short intervals
  6. Case 1: H small.
  7. Case 2: H large
  8. Proof of Theorem \ref{['thm: Steinhaus']}
  9. Concluding remarks

Key Result

Theorem 1.2

Let integer $X$ be large and $W(X)$ tend to infinity arbitrarily slowly as $X$ tends to infinity. Let $H: = H(X) \ll X(\log X)^{-W(X)}$ and $H\to +\infty$ as $X \to +\infty$. Then, for almost all $f\in \mathcal{F}_{X+H}$, as $X\to +\infty$, where $x$ is chosen uniformly from $[X]$.

Theorems & Definitions (33)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • proof : Proof of Theorem \ref{['thm: main']}, assuming Theorem \ref{['thm: Steinhaus']}
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 23 more