Diophantine FLINT-HILLS series

Nikos Mantzakouras; Carlos López Zapata

doi:10.13140/rg.2.2.13412.90240

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arXiv · 2502.03474

Diophantine FLINT-HILLS series

Nikos Mantzakouras·Carlos López Zapata

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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.

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Nikos Mantzakouras, Carlos López Zapata. 2025-01-21. Diophantine FLINT-HILLS series. https://doi.org/10.13140/rg.2.2.13412.90240

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