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The simple normality of the fractional powers of two and the Riemann zeta function

Yuya Kanado, Kota Saito

Abstract

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on arithmetic progressions. As a consequence, we show that \[ \lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } ζ\left(\frac{2 nπi}{\log 2}\right) \frac{e^{2nπi p/q} }{n} =0 \] if and only if $2^{p/q}$ is simply normal to base $2$.

The simple normality of the fractional powers of two and the Riemann zeta function

Abstract

A real number is called simply normal to base if its base- expansion has each digit appearing with average frequency tending to . In this article, we discover a relation between the frequency that the digit appears in the binary expansion of and a mean value of the Riemann zeta function on arithmetic progressions. As a consequence, we show that if and only if is simply normal to base .
Paper Structure (7 sections, 26 theorems, 149 equations)

This paper contains 7 sections, 26 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. A Preliminary discussion
  3. Outline of the proof of Theorem \ref{['theorem:generalization']}
  4. Applying Perron's formula and the residue theorem
  5. Evaluation of upper bounds for $R$ and $S$
  6. Applying the functional equation to $S$ and the proof of Theorem \ref{['theorem:main2']}
  7. Ridout's theorem and the completion of the proof

Key Result

Theorem 1.1

Let $p$ and $q$ be relatively prime integers with $1\leq p<q$. Then we have where $l$ runs over positive real numbers. Especially, we have if and only if $2^{p/q}$ is simply normal to base $2$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['theorem:main1']} assuming Theorem \ref{['theorem:generalization']}
  • Lemma 2.5
  • ...and 43 more