The Borel transform and linear nonlocal equations: applications to zeta-nonlocal field models

We define rigorously operators of the form $f({\mathrm{\partial }}_t)$, in which $f$ is an analytic function on a simply connected domain. Our formalism is based on the Borel transform on entire functions of exponential type. We study existence and regularity of real-valued solutions for the nonlocal in time equation \begin{equation*} f(\partial_t) \phi = J(t) ; ; , \quad t\in $\mathbb{R}$; , \end{equation*}. and we find its more general solution as a restriction to $\mathbb{R}$ of an entire function of exponential type. As an important special case, we solve explicitly the linear nonlocal zeta field equation \begin{equation*} \zeta(\partial_t^2+h)\phi = J(t); , \end{equation*} in which $h$ is a real parameter, $\zeta$ is the Riemann zeta function, and $J$ is an entire function of exponential type. We also analyze the case in which $J$ is a more general analytic function (subject to some weak technical assumptions). This case turns out to be rather delicate: we need to re-interpret the symbol $\zeta ({\mathrm{\partial }}_t^2+h)$ and to leave the class of functions of exponential type. We prove that in this case the zeta-nonlocal equation above admits an analytic solution on a Runge domain determined by $J$. The linear zeta field equation is a linear version of a field model depending on the Riemann zeta function arising from $p$-adic string theory.