Bernoulli Operator and Riemann's Zeta Function

2010/11/15 by Yu, Yiping
#11M06 #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1011.3352

Abstract

We introduce a Bernoulli operator,let B denote the operator symbol,for n=0,1,2,3,... let Bⁿ: = B_n (where B_n are Bernoulli numbers,B₀ = 1,B₁ = 1/2,B₂ = 1/6,B₃ = 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 B^{1 - s} = ζ(s)(s - 1), γ= - log B,where γ is Euler constant.Moreover,we obtain an analogue of the Riemann Hypothesis (All zeros of the function ξ(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 ξ(B + s) has infinitely many zeros on the imaginary axis). \par In addition,we obtain a functional equation of log Π(Bs) and a functional equation of log ζ(B + s) by using Bernoulli operator.

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