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We define rigorously operators of the form $f(\partial_t)$, in which $f$ isan analytic function on a simply connected domain. Our formalism is based onthe Borel transform on entire functions of exponential type. We study existenceand 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 zetafield equation \begin{equation*} \zeta(\partial_t^2+h)\phi = J(t)\; ,\end{equation*} in which $h$ is a real parameter, $\zeta$ is the Riemann zetafunction, and $J$ is an entire function of exponential type. We also analyzethe case in which $J$ is a more general analytic function (subject to some weaktechnical assumptions). This case turns out to be rather delicate: we need tore-interpret the symbol $\zeta(\partial_t^2+h)$ and to leave the class offunctions of exponential type. We prove that in this case the zeta-nonlocalequation 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 dependingon the Riemann zeta function arising from $p$-adic string theory.

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