Amplitudes and the Riemann Zeta Function

TL;DR

This paper explores the connection between scattering amplitudes and the Riemann zeta function, linking physical properties like unitarity and locality to the zeros and meromorphicity of the zeta function, offering a novel physical perspective on its properties.

Contribution

It constructs a closed-form scattering amplitude model based on the zeros of the Riemann zeta function, connecting physical principles to the Riemann hypothesis and properties of the zeta function.

Findings

Amplitude describes exchange of a tower of states with masses related to zeta zeros.

Requiring real masses aligns with the Riemann hypothesis.

Unitarity bounds imply positivity conditions on moments of inverse squared masses.

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 $m_n^2 = \mu_n^2$, where $\zeta\left(\frac{1}{2} \pm i\mu_n\right) = 0$. 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 $\zeta$. Unitarity bounds from dispersion relations for the forward amplitude translate to positivity of the odd moments of the sequence of $1/\mu_n^2$.