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

Michaela Cully-Hugill, Daniel R. Johnston

Abstract

We give an explicit $O(x/T)$ error term for the truncated Riemann--von Mangoldt explicit formula. For large $x$, this provides a modest improvement over previous work, which we demonstrate via an application to a result on primes between consecutive powers.

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

Abstract

We give an explicit error term for the truncated Riemann--von Mangoldt explicit formula. For large , this provides a modest improvement over previous work, which we demonstrate via an application to a result on primes between consecutive powers.
Paper Structure (8 sections, 14 theorems, 95 equations, 2 tables)

This paper contains 8 sections, 14 theorems, 95 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Outline of paper
  3. An averaged, weighted, truncated Perron formula
  4. A modified Perron formula
  5. Proof of theorems \ref{['mainthm']} and \ref{['mainthm-intervals']}
  6. Application: primes between consecutive powers
  7. Current zero-free regions
  8. Updating bounds for an integral over $\zeta'/\zeta$

Key Result

Theorem 1.1

For any $\alpha\in(0,1/2]$ there exist constants $M$ and $x_M$ such that for $\max\{51,\log x\}<T<(x^{\alpha}-2)/2$, for all $x\geq x_M$. Some admissible values of $x_M$, $\alpha$ and $M$ are $(40,1/2,5.03)$ and $(10^3, 1/100, 0.5597)$, with more given in CH_DJ_Perron1.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • Remark
  • proof
  • proof : Proof of Theorem \ref{['mainramthm']}
  • Lemma 3.1
  • ...and 11 more