The $p$-adic sector of the adelic string

We study the construction of Lagrangians that can be considered the Lagrangians of the $p$-adic sector of an adelic open scalar string. Such Lagrangians are closely related to the Lagrangian for a single $p$-adic string and contain the Riemann zeta function with the d'Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach that combines all $p$-adic Lagrangians. This new Lagrangian is attractive because it is an analytic function of the d'Alembertian. Investigating the field theory with the Riemann zeta function is also interesting in itself.