On error sums formed by rational approximations with split denominators

TL;DR

This paper studies specialized error sums involving rational approximations with split denominators for various fundamental constants, revealing connections to known integer sequences and generalizing classical error sum theory.

Contribution

It introduces a generalized framework for error sums with split denominators, extending classical error sum concepts and linking them to special constants and integer sequences.

Findings

Error sums for constants like π, e, and ζ(2), ζ(3) are explicitly analyzed.

Connections between rational coefficients in error sums and well-known integer sequences are established.

The theory generalizes traditional error sums based on continued fraction convergents.

Abstract

In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_m\alpha - \frac{a_m}{c_m}\,\Big) \,,\] where $\alpha$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m ={(-1)}^m$. In particular, we investigate such sums for \[\alpha \in \big\{ \pi, e,e^{1/2},e^{1/3},\dots, \log (1+t), \zeta(2), \zeta(3) \big\} \] and exhibit some connections between rational coefficients occurring in error sums for Ap\'ery's continued fraction for $\zeta(2)$ and well-known integer sequences. The concept of the paper generalizes the theory of ordinary error sums, which are given by $b_m=q_m$ and $a_m/c_m=p_m$ with the convergents $p_m/q_m$ from the continued fraction expansion of $\alpha$.