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.
Categories: math.NT math.CV