黎曼ζ-函数之一:KS-变换——献给杨乐先生八十华诞

On the Riemann Zeta Function,Ⅰ: KS-transform

我们定义了KS-变换和自然数乘法结构相关的Fourier变换,建立了实数乘法半群[1,∞)={x:x∈R,x≥1}和复半平面Ω={s=σ+it:σ,t∈R,σ≥1/2}之间的由KS-变换诱导的对偶关系,证明了KS-变换是希尔伯特空间L^2([1,∞))和哈代空间H^2(Ω)之间的等距算子,而且该算子保持了相关的函数空间之间由实数的乘法卷积和复数点点相乘诱导出的代数结构的同构.作为应用,我们给出了黎曼假设成立的有关算子指标的等价命题,从而算子理论为研究黎曼ζ-函数和自然数的乘法结构提供了新思路.

Kadison-Singer transform(KS-transform)is introduced as a multiplicative Fourier transform associated with the multiplicative structure of natural numbers.It is a unitary operator between the Hilbert space L2([1,∞))and Hardy space H^2(Ω),whereΩis a the right half complex plane with the real part great than or equal to 1/2.We also show that KS-transform maps the multiplicative convolution of two functions on[1,∞)to the usual product of functions onΩ.Riemann hypothesis is equivalent to the vanishing index of certain convolution operators.