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On the variance of squarefree integers in short intervals and arithmetic progressions

Ofir Gorodetsky, Kaisa Matomäki, Maksym Radziwiłł, Brad Rodgers

Abstract

We evaluate asymptotically the variance of the number of squarefree integers up to $x$ in short intervals of length $H < x^{6/11 - \varepsilon}$ and the variance of the number of squarefree integers up to $x$ in arithmetic progressions modulo $q$ with $q > x^{5/11 + \varepsilon}$. On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $H < x^{2/3 - \varepsilon}$ and $q > x^{1/3 + \varepsilon}$. Furthermore we show that obtaining a bound sharp up to factors of $H^{\varepsilon}$ in the full range $H < x^{1 - \varepsilon}$ is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

On the variance of squarefree integers in short intervals and arithmetic progressions

Abstract

We evaluate asymptotically the variance of the number of squarefree integers up to in short intervals of length and the variance of the number of squarefree integers up to in arithmetic progressions modulo with . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively and . Furthermore we show that obtaining a bound sharp up to factors of in the full range is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

Paper Structure

This paper contains 25 sections, 23 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Fractional Brownian motion
  4. Acknowledgments
  5. Conventions and Notations
  6. Proofs of Theorems \ref{['thm:main']} and \ref{['thm:main2']}
  7. Lemmas
  8. Dirichlet polynomials and $L$-functions
  9. Asymptotic estimates
  10. Initial reductions on second moments
  11. Point-counting lemmas
  12. The range $H^{1+\varepsilon} \leq z \leq \min\{X/H^{1/2+\varepsilon}, H^{1/2-\varepsilon}X^{1/2}\}$ in the $t$-aspect : Proof of Proposition \ref{['pr:prop1']}
  13. The range $z \geq H^{4/3+\varepsilon}$ in the $t$-aspect : Proof of Proposition \ref{['pr:prop2']}
  14. The range $(x/q)^{1+\varepsilon} \leq z < x^{-\varepsilon} \sqrt{qx}$ in the $q$-aspect : Proof of Proposition \ref{['prop:prop1q']}
  15. Proof of Proposition \ref{['prop:asymptqaspect']}
  16. ...and 10 more sections

Key Result

Theorem 1

Let $\varepsilon \in (0, \tfrac{1}{100})$ be given. Let $X \geq 1$ and $1 \le H \leq X^{6/11 - \varepsilon}$. Then with Assuming the Lindelöf Hypothesis eq:main holds in the wider range $H \leq X^{2/3-\varepsilon}$.

Figures (1)

  • Figure 1: Partial sums of $\mu^2(n)$ : depiction of \ref{['eq:broww']} with $x = 2 \times 10^{15}$, $H = 44721359$ and $0 \leq t \leq 10$.

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2
  • proof : Proof of Theorem \ref{['thm:main']} assuming Proposition \ref{['pr:prop1']} and Proposition \ref{['pr:prop2']}
  • Proposition 3
  • Proposition 4
  • Lemma 1: Large-value theorem
  • proof
  • ...and 27 more