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Notes On An Approach To Apery's Constant

Leon D. Fairbanks

Abstract

The Basel problem, solved by Leonhard Euler in 1734, asks to resolve $ζ(2)$, the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series: \begin{equation} \sum_{i=1}^{\infty}\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots\notag \end{equation} The same question is posed regarding the summation of the reciprocals of the cubes of the natural numbers, $ζ(3)$. The resulting constant is known as Apery's constant. A YouTube channel, 3BlueBrown, produced a video entitled, "Why is pi here? And why is it squared? A geometric answer to the Basel problem". The video presents the work of John Wästlund. The equations can be extended to $ζ(n)$, but the geometric argument is lost. We try to explore these equations for $ζ(n)$.

Notes On An Approach To Apery's Constant

Abstract

The Basel problem, solved by Leonhard Euler in 1734, asks to resolve , the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series: \begin{equation} \sum_{i=1}^{\infty}\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots\notag \end{equation} The same question is posed regarding the summation of the reciprocals of the cubes of the natural numbers, . The resulting constant is known as Apery's constant. A YouTube channel, 3BlueBrown, produced a video entitled, "Why is pi here? And why is it squared? A geometric answer to the Basel problem". The video presents the work of John Wästlund. The equations can be extended to , but the geometric argument is lost. We try to explore these equations for .
Paper Structure (14 sections, 129 theorems, 428 equations)

This paper contains 14 sections, 129 theorems, 428 equations.

Table of Contents

  1. Introduction
  2. Zeta At Even Integer Values
  3. Zeta(3)
  4. Zeta At Odd Integer Values
  5. Some Trig Integrals
  6. Formulas Involving $\frac{(n!)^2}{(n+j)!(n-j)!}$
  7. Dynamic Binomial Sums And The Difference Operator
  8. More Zeta Series And Trig Integrals
  9. Related Functions
  10. Nested Root Function Identities
  11. A More Direct Proof Of Zeta Series Results
  12. Infinite Linear Systems Of Equations
  13. Partial List Of Results
  14. Bibliography

Key Result

Lemma 1.1

Assume $m\in\mathbb{Z}$, $m>0$ then

Theorems & Definitions (247)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • ...and 237 more