A Continued-Fraction Criterion for the Flint Hills Series, and the Integer Sequences It Generates

Abstract

The Flint Hills series is the sum of the reciprocals of the cube of an integer times the square of the sine of that integer, taken over the positive integers, and whether it converges is a well-known open problem, tied to the irrationality measure of pi. We show that its convergence is governed entirely by an integer sequence, namely the sequence of partial quotients of the continued fraction of the reciprocal of pi. We prove that the series converges if and only if the sum of the squares of consecutive partial quotients, each divided by the corresponding convergent denominator, is finite. This transforms an analytic problem into an explicit statement about the growth of a single integer sequence, already catalogued by Sloane, and about the sequence of convergent denominators. We isolate the finite integer spikes produced by the exceptionally large partial quotient at position five, we describe the indices at which such spikes occur, and we show, using the Gauss-Kuzmin statistics of partial quotients, that for almost every real number the corresponding series converges. The reciprocal of pi is thus a single arithmetic point inside a set of measure zero.