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On the error in the prime number theorem in short intervals

Ethan Simpson Lee

Abstract

Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both constant terms and applicable ranges, notably improving and extending earlier work by Cully-Hugill and Dudek.

On the error in the prime number theorem in short intervals

Abstract

Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both constant terms and applicable ranges, notably improving and extending earlier work by Cully-Hugill and Dudek.
Paper Structure (10 sections, 11 theorems, 74 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Explicit approximation for a weighted sum
  3. Explicit approximation for a weighted sum
  4. Step 1: The inverse Mellin transform
  5. Step 2: An explicit formula
  6. Step 3: A parametrised approximate formula
  7. Step 4: Final deductions
  8. Proof of main results
  9. Proof of Corollary \ref{['cor:main_pnt']}
  10. Proof of Corollary \ref{['cor:main_mt']}

Key Result

Theorem 1.1

Suppose that the RH is true, $x \geq e^{20}$, and If $\sqrt{x}\log{x} \leq h \leq x^{3/4}$, then

Figures (1)

  • Figure 1: The weight function $w_{\pm}(t)$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark
  • Theorem 2.1
  • Remark
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 16 more