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On the error term in the explicit formula of Riemann--von Mangoldt

Michaela Cully-Hugill, Daniel R. Johnston

Abstract

We provide an explicit $O(x\log x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramaré (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.

On the error term in the explicit formula of Riemann--von Mangoldt

Abstract

We provide an explicit error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramaré (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.
Paper Structure (10 sections, 18 theorems, 101 equations, 4 tables)

This paper contains 10 sections, 18 theorems, 101 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of main results
  4. Outline of paper
  5. An explicit truncated Perron formula
  6. Zero-free regions
  7. Wolke's method
  8. Proof of Theorem \ref{['mainthm']}
  9. Application: primes between consecutive powers
  10. Application: the error term in the prime number theorem

Key Result

Theorem 1.1

Take $50<T<x$ for half an odd integer $x>e^{60}$. Then

Theorems & Definitions (31)

  • Theorem 1.1: Dudek
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:new-main']}
  • Lemma 3.1: P_T-RH_21
  • ...and 21 more