2022-1-23 21:54

Figure 1 Illustration of A(s) defined in Eq. (3). As shown in text, A(s) is meromorphic, with poles at s=μn2 corresponding to the nontrivial zeros of the Riemann zeta function, ζ(1/2±iμn)=0.

πeφ这样的字母鸡′′′经常出现在科学和数学中意想不到的地方。帕斯卡三角形（Pascal's triangle）和斐波那契数列（Fibonacci sequence）似乎也在自然界中普遍存在。还有黎曼ζ函数，一个看似简单的函数，自19世纪以来一直困扰着数学家。最著名的难题是黎曼假设（Riemann hypothesis），它可能是数学中最大的未解问题。克雷数学研究所(Clay mathematics Institute)悬赏100万美元，奖励能正确证明黎曼假设的人。

1/1x+1/2x+1/3x+······

1859年， 波恩哈德·黎曼（Bernhard Riemann）决定考虑当x是复数时会发生什么。这个函数现在被命名为黎曼ζ函数（Riemann zeta），它输入一个复数，然后输出另一个复数。

Abstract

Physical properties of scattering amplitudes are mapped to the Riemann zeta function. Specifically, a closed-form amplitude is constructed, describing the tree-level exchange of a tower with masses mn2=μn2, where ζ(1/2±n)=0. Requiring real masses corresponds to the Riemann hypothesis, locality of the amplitude to meromorphicity of the zeta function, and universal coupling between massive and massless states to simplicity of the zeros of ζ. Unitarity bounds from dispersion relations for the forward amplitude translate to positivity of the odd moments of the sequence of 1/μn2.