Amplitudes and the Riemann Zeta Function

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 $\zeta (1/2\pm i{\mu }_n)=0$. 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 $\zeta$. Unitarity bounds from dispersion relations for the forward amplitude translate to positivity of the odd moments of the sequence of $1/{\mu }_n^2$.

Figure 1

Illustration of $\mathcal{A}(s)$ defined in Eq. (3). As shown in text, $\mathcal{A}(s)$ is meromorphic, with poles at $s={\mu }_n^2$ corresponding to the nontrivial zeros of the Riemann zeta function, $\zeta (1/2\pm i{\mu }_n)=0$.

Figure 2

Taylor series of $\mathcal{M}(s,0)$ near $s=0$. All even derivatives of $\mathcal{M}(s,0)$ at $s=0$ are positive, as required by analyticity and unitarity for a forward amplitude or alternatively by the Riemann hypothesis.