Zeta-nonlocal scalar fields

We consider some nonlocal and nonpolynomial scalar field models originating from $p$-adic string theory. An infinite number of space-time derivatives is determined by the operator-valued Riemann zeta function through the d'Alembertian $\Box$ in its argument. The construction of the corresponding Lagrangians $L$ starts with the exact Lagrangian $\mathcal L_p$ for the effective field of the $p$-adic tachyon string, which is generalized by replacing $p$ with an arbitrary natural number $n$ and then summing $\mathcal L_n$ over all $n$. We obtain several basic classical properties of these fields. In particular, we study some solutions of the equations of motion and their tachyon spectra. The field theory with Riemann zeta-function dynamics is also interesting in itself.