Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smooth Numbers in Short Intervals

Sarvagya Jain

Abstract

Let \( X \geq y \geq 2 \), and let \( u = \frac{\log X}{\log y} \). We say a number is \textit{$y$-smooth} if all of its prime factors are less than or equal to \( y \). In this paper, we study the distribution of $y$-smooth numbers in short intervals. In particular, for \( y \geq \exp\left( (\log X)^{2/3 + ε} \right) \), we show that the interval \( [x, x+h] \) contains a $y$-smooth number for almost all \( x \in [X, 2X] \), provided \( h \geq \exp\left( (1 + ε) \left( \frac{11}{8} u \log u + 4 \log \log X \right) \right) \), and \( X \) is sufficiently large depending on \( ε\). This result improves upon an earlier result by Matomäki. Additionally, we provide the corresponding ``all intervals" type result.

Smooth Numbers in Short Intervals

Abstract

Let , and let . We say a number is \textit{-smooth} if all of its prime factors are less than or equal to . In this paper, we study the distribution of -smooth numbers in short intervals. In particular, for \( y \geq \exp\left( (\log X)^{2/3 + ε} \right) \), we show that the interval contains a -smooth number for almost all , provided \( h \geq \exp\left( (1 + ε) \left( \frac{11}{8} u \log u + 4 \log \log X \right) \right) \), and is sufficiently large depending on . This result improves upon an earlier result by Matomäki. Additionally, we provide the corresponding ``all intervals" type result.

Paper Structure

This paper contains 13 sections, 20 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Sketch of the Argument
  3. Acknowledgements
  4. Dirichlet Polynomial Estimates
  5. Some Results About Smooth Numbers
  6. Defining the Dirichlet Polynomial
  7. Parseval Reduction
  8. Bounding the Mean Value
  9. Contribution from T1
  10. Contribution from T2
  11. Contribution from T3
  12. Finishing the Proof of Theorem \ref{['almost all interval']}
  13. Existence in All Intervals

Key Result

Theorem 1.1

For any $\varepsilon > 0$, there exists a positive constant $C = C(\varepsilon)$ such that the following holds. Let $X \geq 2$ be large enough depending on $\varepsilon$. If and for $u = \frac{\log X}{\log y}$, then the interval $[x, x+h]$ contains a $y$-smooth number for almost all $x \in [X, 2X]$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 33 more