Modified equations and the Basel problem

TL;DR

This paper explores the connection between modified equations in numerical discretizations and the Basel problem, deriving a series expansion that confirms the value of the Riemann zeta function at 2.

Contribution

It provides a novel derivation linking modified equations of discretized harmonic oscillators to the Basel problem, illustrating a new approach to classical mathematical series.

Findings

Derived a series expansion for (arcsin(h/2))^2

Connected modified equations to the Basel problem

Confirmed that ζ(2) = π^2/6

Abstract

Discretizations of differential equations are often studied through their modified equation. This is a differential equation, usually obtained as a power series, with solutions that exactly interpolate the discretization. By comparing the St\"ormer-Verlet discretization of the harmonic oscillator with its modified equation, we obtain a relatively simple derivation of the expansion \[ \left( \arcsin \frac{h}{2} \right)^2 = \frac{1}{2} \sum_{k=1}^\infty \frac{(k-1)!^2}{(2k)!} h^{2k}, \] which can be used to show that $\zeta(2) = \frac{\pi^2}{6}$.