Thirty-two Goldbach Variations

Abstract

We give thirty-two diverse proofs of a small mathematical gem--the fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating both the wide variety of techniques fruitfully used to study such sums and the attraction of their study.

v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory material added and material on inequalities, Hilbert matrix and Witten zeta functions. Errors in the second section on Complex Line Integrals are corrected. To appear in International Journal of Number Theory. Title changed

References in corpus (7)

• Multiple $q$-Zeta Values
• Parametric Euler Sum Identities
• The Algebra and Combinatorics of Shuffles and Multiple Zeta Values
• Duality for Finite Multiple Harmonic q-Series
• A q-analog of Euler's decomposition formula for the double zeta function
• Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth
• Partition Identities for the Multiple Zeta Function

Cited by in corpus (1)

• On a family of polynomials related to $\zeta (2,1)=\zeta (3)$