Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bernoulli Operator and Riemann's Zeta Function   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.

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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom