  Some nonlocal and nonpolynomial scalar field models originated from p-adic
string theory are considered. Infinite number of spacetime derivatives is
governed by the Riemann zeta function through d'Alembertian $\Box$ in its
argument. Construction of the corresponding Lagrangians begins with the exact
Lagrangian for effective field of p-adic tachyon string, which is generalized
replacing p by arbitrary natural number n and then taken a sum of over all n.
Some basic classical field properties of these scalar fields are obtained. In
particular, some trivial solutions of the equations of motion and their tachyon
spectra are presented. Field theory with Riemann zeta function nonlocality is
also interesting in its own right.
