Bernoulli Operator and Riemann's Zeta Function

TL;DR

This paper introduces a Bernoulli operator to derive new formulas for the Riemann zeta function, Euler constant, and related number-theoretic functions, proposing analogues of the Riemann Hypothesis and Hardy's theorem.

Contribution

It defines a Bernoulli operator and uses it to establish novel formulas and hypotheses related to the Riemann zeta function and other L-functions.

Findings

Derived formulas linking Bernoulli operator to zeta function and Euler constant

Proposed an analogue of the Riemann Hypothesis involving Bernoulli operator

Established functional equations using Bernoulli operator

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.