Diophantine FLINT-HILLS series

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.