  We define rigorously operators of the form $f(\partial_t)$, 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 $\zeta(\partial_t^2+h)$ 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.
