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The variance of integers without small prime factors in short intervals

Ofir Gorodetsky

Abstract

The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the variance of integers without prime factors below $y$, in short intervals. We use complex analysis and sieve theory to prove an unconditional asymptotic result in a range for which we give evidence is qualitatively best possible. We find that this variance connects with statistics of $y$-smooth numbers, and, as with primes, is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of $y$.

The variance of integers without small prime factors in short intervals

Abstract

The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the variance of integers without prime factors below , in short intervals. We use complex analysis and sieve theory to prove an unconditional asymptotic result in a range for which we give evidence is qualitatively best possible. We find that this variance connects with statistics of -smooth numbers, and, as with primes, is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of .

Paper Structure

This paper contains 23 sections, 24 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. The variance of primes in short intervals
  3. Integers without small prime factors
  4. Main result
  5. Estimates of the main term
  6. Structure of paper
  7. Discussion
  8. The case of X being a primorial
  9. Fluctuations and limitations
  10. Duality
  11. Conventions
  12. Application of sieve theory
  13. Main term estimates: large y
  14. Main term estimates: small y
  15. Hall's identities
  16. ...and 8 more sections

Key Result

Theorem 1.1

Fix $\varepsilon\in (0,1)$. Suppose the parameters $y,H\le X$ tend to $\infty$ in such a way that $y \ge (2+\varepsilon)\log H$ holds. Set $a=\log \log H/\log y \in (0, 1)$. If then, as $H\to \infty$,

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Conjecture 1
  • Theorem 1.3
  • Remark 1
  • Remark 2
  • Lemma 1.4
  • Lemma 1.5
  • proof
  • Conjecture 2
  • ...and 33 more