Bernoulli Operator and Riemann's Zeta Function

• Published in: arXiv.org 2015-09
• Main Author: Yu, Yiping
• Format: Article
• Language: English
• Subjects: Dirichlet problem; Functional equations; Operators (mathematics)
• ISSN: 2331-8422

We introduce a Bernoulli operator,let \(\mathbf{B}\) denote the operator symbol,for n=0,1,2,3,... let \({\mathbf{B}^n}: = {B_n}\) (where \({B_n}\) are Bernoulli numbers,\({B_0} = 1,B{}_1 = 1/2,{B_2} = 1/6,{B_3} = 0\)...).We obtain some formulas for Riemann's Zeta function,Euler constant and a number-theoretic function relate to Bernoulli operator.For example,we show that \[{\mathbf{B}^{1 - s}} = \zeta (s)(s - 1),\] \[\gamma = - \log \mathbf{B},\]where \({\gamma}\) is Euler constant.Moreover,we obtain an analogue of the Riemann Hypothesis (All zeros of the function \(\xi (\mathbf{B} + s)\) lie on the imaginary axis).This hypothesis can be generalized to Dirichlet L-functions,Dedekind Zeta function,etc.In particular,we obtain an analogue of Hardy's theorem(The function \(\xi (\mathbf{B} + s)\) has infinitely many zeros on the imaginary axis). \par In addition,we obtain a functional equation of \(\log \Pi (\mathbf{B}s)\) and a functional equation of \(\log \zeta (\mathbf{B} + s)\) by using Bernoulli operator.