Bessel Averaging, Fourier Decomposition, and the Value of the Borwein-Bailey-Girgensohn Series

TL;DR

This paper analyzes the Borwein--Bailey--Girgensohn sinusoidal series, decomposing it into a Bessel averaging part and a remainder, evaluates the main term exactly, and provides a conjecture for its sum based on numerical evidence.

Contribution

It introduces a novel decomposition of the series into a Bessel averaging series and a Fourier harmonic remainder, and evaluates the main term explicitly.

Findings

Main term equals log 6

Series sum approximates li(3)

Conjecture reduces to a Diophantine identity

Abstract

We study the Borwein--Bailey--Girgensohn sinusoidal series S_BBG = sum_{n=1}^\infty (1/n) * ((2+sin n)/3)^n, originally posed as an open problem by Borwein, Bailey, and Girgensohn, whose convergence was established by Boppana using the irrationality measure of pi. We present three unconditional results. First, applying the Weyl equidistribution theorem with a quantitative Erdos--Turan bound, we split S_BBG = M + R, where M = sum_{n=1}^\infty I_n/n is a Bessel averaging series and |R| < infinity. Second, we evaluate M exactly via Fubini's theorem and the Fourier series of log(1-cos t): M = sum_{n=1}^\infty I_n/n = log 6. Third, we decompose the remainder R into a convergent series of Fourier harmonics: R = sum_{k=1}^\infty 2*Re[G_k((2/3)e^{ik})], where each G_k(z) = sum_{n=1}^\infty c_k(n) z^n/n is a Dirichlet-type generating function built from the k-th Fourier coefficients of (theta -> (1 + (sin theta)/2)^n). The series converges absolutely because |(2e^{ik})/3| = 2/3 < 1. Numerical computation strongly suggests S_BBG = Ei(log 3) = li(3) approx 2.16358...; we reduce this conjecture to a single Diophantine identity for R and indicate the Mellin-transform approach most likely to settle it.