Diophantine FLINT-HILLS series

TL;DR

This paper proves the convergence of the FLINT-HILLS series, introduces new criteria for similar diophantine series, and links it to the Fermi-Dirac integral, providing insights into the irrationality measure of pi.

Contribution

It establishes convergence criteria for the FLINT-HILLS series and connects it to other mathematical functions, offering new bounds on pi's irrationality measure.

Findings

Proved convergence of the FLINT-HILLS series.

Connected the series to the Fermi-Dirac integral.

Suggested upper bound of pi's irrationality measure is 2.5.

Abstract

We prove the convergence of the FLINT-HILLS series and establish new criteria for a similar type of diophantine or lacunary series, which faces issues due to spaced long terms coming from the trigonometric nature of functions, e.g., cosecant in the FLINT-HILLS series. We connect the FLINT-HILLS series to the Fermi-Dirac integral via the Riemann-Stieltjes integral and YOUNG'S inequality criteria but also proved that the upper bound of the irrationality measure of pi is equal or lower than 2.5 expected if the FLINT-HILLS series converged.