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Worse than square-root cancellation in Bateman-Horn's conjecture

Giacomo Bortolussi

Abstract

We prove asymptotics for the average error term in Bateman-Horn's conjecture in the exponential range.

Worse than square-root cancellation in Bateman-Horn's conjecture

Abstract

We prove asymptotics for the average error term in Bateman-Horn's conjecture in the exponential range.

Paper Structure

This paper contains 10 sections, 20 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. The term $\sum {(\psi^{\mathrm{abs}}_P)}^2$
  4. The diagonal term
  5. The non-diagonal term
  6. The term $\sum_P{\mathfrak S_P}^2$
  7. Hooley's neutralisers
  8. Calculations for $\sum_{P}\mathfrak S_P^2$
  9. The term $\sum \psi^{\mathrm{abs}}_P\mathfrak S_P$
  10. Transition from $\psi^{\mathrm{abs}}$ to $\psi$

Key Result

Theorem 1.1

Fix $d\ge 2$ and $\delta\ge1$. Then for all $x,H>2$ with $(\log H)\le x\le (\log H)^\delta$ we have where the implied constant depends only on $d$ and $\delta$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['teo_mainabs']}
  • Lemma 3.1
  • proof
  • ...and 25 more