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Polylogarithmic Decomposition of a Borwein--Bailey--Girgensohn Series and its Connection to Ei(log 3)

Carlos Lopez Zapata

Abstract

We study the Borwein-Bailey-Girgensohn sinusoidal series S = \sum_{n=1}^{\infty} (1/n) ((2+sin n)/3)^n, originally posed as an open problem by Borwein, Bailey, and Girgensohn (2004). Its convergence was recently established by Boppana (2020) using the irrationality measure of pi, yet the exact value remained elusive; partial sums to 10^7 terms suggest S \approx 2.163. We make three primary contributions. First, by introducing the generating function f(x) = (1/x)((2+sin x)/3)^x, we derive an exact polylogarithmic decomposition S = log 3 - (1/2)log 2 + \sum_{m=1}^{\infty} λ_m Li_{-m}(1/3) + E, where the coefficients λ_m satisfy |λ_m| \leq (log 2)^m/m! and E is a finite error term. Second, applying the Weyl equidistribution theorem with a quantitative Erdos-Turan bound, we decompose the series into S = M + R, where M is the Bessel averaging series and R is a controlled Diophantine remainder. Third, we evaluate the averaging term exactly as M = log 6. As a consequence, we show that the identification of the series value with the exponential integral Ei(log 3) is equivalent to the evaluation of the remainder as R = Ei(log 3) - log 6 \approx 0.3718, thereby reducing the open problem to a precise Diophantine target.

Polylogarithmic Decomposition of a Borwein--Bailey--Girgensohn Series and its Connection to Ei(log 3)

Abstract

We study the Borwein-Bailey-Girgensohn sinusoidal series S = \sum_{n=1}^{\infty} (1/n) ((2+sin n)/3)^n, originally posed as an open problem by Borwein, Bailey, and Girgensohn (2004). Its convergence was recently established by Boppana (2020) using the irrationality measure of pi, yet the exact value remained elusive; partial sums to 10^7 terms suggest S \approx 2.163. We make three primary contributions. First, by introducing the generating function f(x) = (1/x)((2+sin x)/3)^x, we derive an exact polylogarithmic decomposition S = log 3 - (1/2)log 2 + \sum_{m=1}^{\infty} λ_m Li_{-m}(1/3) + E, where the coefficients λ_m satisfy |λ_m| \leq (log 2)^m/m! and E is a finite error term. Second, applying the Weyl equidistribution theorem with a quantitative Erdos-Turan bound, we decompose the series into S = M + R, where M is the Bessel averaging series and R is a controlled Diophantine remainder. Third, we evaluate the averaging term exactly as M = log 6. As a consequence, we show that the identification of the series value with the exponential integral Ei(log 3) is equivalent to the evaluation of the remainder as R = Ei(log 3) - log 6 \approx 0.3718, thereby reducing the open problem to a precise Diophantine target.
Paper Structure (25 sections, 9 theorems, 30 equations)

This paper contains 25 sections, 9 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. History of the problem
  3. Boppana's convergence theorem
  4. The conjecture and our contributions
  5. Notation and conventions
  6. The Generating Function and Its Expansion
  7. The generating function
  8. Logarithmic factorisation
  9. Expansion of the exponential factor
  10. The Polylogarithmic Decomposition
  11. Polylogarithm generating sums
  12. The decomposition theorem
  13. The Weyl Equidistribution Decomposition
  14. Setup
  15. The Bessel Averaging Series and Its Exact Value
  16. ...and 10 more sections

Key Result

Theorem 1.1

The series $S_{\mathrm{BBG}}=\sum_{n=1}^{\infty}\frac{1}{n} \bigl(\frac{2+\sin n}{3}\bigr)^n$ converges.

Theorems & Definitions (27)

  • Theorem 1.1: Boppana Boppana2020
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Logarithmic factorisation
  • proof
  • Lemma 2.4: Exponential expansion
  • Lemma 3.1: Standard generating sums
  • proof
  • Theorem 3.2: Polylogarithmic decomposition
  • proof
  • ...and 17 more