1994/01/01 by Richard E. Crandall, Joe Buhler · 2 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Arithmetic zeta function #Backward Euler method #Euler equations #Euler number (physics) #Euler summation #Euler's formula #Euler's totient function #Function (biology) #Harmonic number #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Prime zeta function #Proof of the Euler product formula for the Riemann zeta function #Pure mathematics #Riemann hypothesis #Riemann zeta function #Semi-implicit Euler method #Series (stratigraphy) #Type (biology)
paper · doi:10.1080/10586458.1994.10504297
openalex publication_date 1994/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Euler studied double sums of the form for positive integers r and s, and inferred, for the special cases r = 1 or r + s odd, elegant identities involving values of the Riemann zeta function. Here we establish various series expansions of ζ(r, s) for real numbers r and s. These expansions generally involve infinitely many zeta values. The series of one type terminate for integers r and s with r + s odd, reducing in those cases to the Euler identities. Series of another type are rapidly convergent and therefore useful in numerical experiments.