Application of the Hurwitz Zeta Function in Many-body Systems with Long-ranged Interactions

**Abstract: **The Hurwitz zeta function $\zeta(s,a)$ is an extension of the Riemann zeta function. In this talk, I will show that the ''reflection rules'' of the Hurwitz zeta function, or equivalently, the polygamma function, can be expressed as the physical problem of a 1D many-body system interacting through a long-ranged pair potential $v(r)=r^{-p}$, where $0\leq p \leq 1$, in a uniform background with the opposite ''charge''. At positive temperatures, these systems lead to unusual ''hyperuniform'' equilibrium configurations characterized by suppressed long-wavelength number density fluctuations, a special case being the Dyson log gas, for which $p=0$. The aforementioned many-body problem can be equivalently studied using particles arranged on a 1D torus, which enables us to define novel metrics on the torus using Hurwitz zeta or polygamma functions.