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On the approximation of the zeta function by Dirichlet polynomials

Juan Arias de Reyna

Abstract

We prove that for $s=σ+it$ with $σ\ge0$ and $0<t\le x$, we have \[ζ(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{(s-1)}+Θ\frac{29}{14} x^{-σ},\qquad \frac{29}{14}=2.07142\dots\] where $Θ$ is a complex number with $|Θ|\le1$. This improves Theorem 4.11 of Titchmarsh.

On the approximation of the zeta function by Dirichlet polynomials

Abstract

We prove that for with and , we have where is a complex number with . This improves Theorem 4.11 of Titchmarsh.

Paper Structure

This paper contains 3 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Explicit versions of some lemmas on exponential sums
  3. Approximation of zeta(s) by a Dirichlet polynomial

Key Result

Lemma \oldthetheorem

Let $f(x)\in L^1[a,b]$ and $g$ be a real positive and decreasing function in the range $[a,b]$. Then there is a number $a\le \xi\le b$ such that

Theorems & Definitions (11)

  • Lemma \oldthetheorem: Bonnet's form of the Second Mean Value Theorem
  • Lemma \oldthetheorem: Explicit version of Lemma 4.3 of Titchmarsh
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem: Explicit version of Lemma 4.7 of Titchmarsh
  • proof
  • Lemma \oldthetheorem: Explicit version of Lemma 4.10 of Titchmarsh
  • proof
  • Theorem \oldthetheorem: Explicit version of Theorem 4.11 in Titchmarsh
  • proof
  • ...and 1 more