1995/06/01 by David Borwein, Jonathan M. Borwein, Roland Girgensohn · 4 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Combinatorial Mathematics #Euler's formula #Goldbach's conjecture #Riemann hypothesis #Mathematics #Mathematical proof #Extrapolation #Pure mathematics #Series (stratigraphy) #Number theory #Discrete mathematics #Mathematical analysis
paper · pdf · doi:10.1017/s0013091500019088
openalex publication_date 1995/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers. As Euler discovered by a process of extrapolation (from s + t ≦ 13), σ h ( s, t ) can be evaluated in terms of Riemann ζ-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we give a formula for the series This evaluation involves ζ-functions and σ h (2, m ).