Bernoulli Operator and Riemann's Zeta Function

Abstract

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.