A new formula for ζ(s)

2018/11/22 by Chenfeng He, He, Chenfeng
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1811.09226

openalex publication_date 2018/11/22 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

In this paper, by introducing a new operation in the vector space of analytic functions, the author presents a method for derivating the well-known formulas: ζ(1-k)=-(B_k)/(k) and ζ(1-n,a)=-(B_n(a))/(n) , where ζ, ζ(1-n,a) denote the Riemann zeta function and the Hurwitz zeta function respectively. B_k is the k-th Bernoulli number. Also the author steps further to deduce some identities related to Bernoulli number and Bernoulli polynomial. Moreover, when combining the operation with forward difference, we can show a new formula for Riemann zeta function, i.e. ζ(s)=e∑_n=0^∞∑_i=0ⁿ(-1)^n-i\frac1(n-i)!(1+i)^s.

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