On the approximation of the zeta function by Dirichlet polynomials

TL;DR

This paper improves the approximation of the Riemann zeta function using Dirichlet polynomials for complex numbers with non-negative real parts, providing explicit error bounds and refining previous results.

Contribution

It provides a sharper bound for approximating the zeta function by Dirichlet polynomials, extending Titchmarsh's theorem with explicit error terms.

Findings

Enhanced approximation accuracy for the zeta function.

Explicit error bounds with a new constant.

Refinement of previous theoretical results.

Abstract

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