A double series for $\pi$ using Fourier series and the Grothendieck-Krivine constant

TL;DR

This paper derives a novel double-series formula for pi using Fourier series and the Parseval identity, revealing a connection to the Grothendieck-Krivine constant and expressing it as a double series.

Contribution

It introduces a new double-series representation for pi involving Fourier analysis and links it to the Grothendieck-Krivine constant, providing a novel mathematical expression.

Findings

Derived a double-series formula for pi using Fourier series.

Connected the Grothendieck-Krivine constant to a double-series expression.

Demonstrated the use of Parseval's identity in deriving series for fundamental constants.

Abstract

We provide a double-series formula for $\pi$ obtained using the Fourier series expansion of $1/\cos(x/4)$ and applying the Parseval-Plancherel identity. We show that such a formula involves the Grothendieck-Krivine constant, and that the latter can therefore be expressed as a double series as well.