From $p$-adic to zeta scalar particles

In this talk, we start from the Lagrangian for p-adic open strings, which is slightly modified so that it does not contain tachyons and ghosts. This Lagrangian makes sense not only for prime numbers but also for all natural numbers. Taking a suitable sum of these Lagrangians over all natural numbers and using Euler’s formula for the Riemann zeta function, a new Lagrangian is obtained. This new Lagrangian contains non-locality through a non-local operator in the form of the Riemann zeta function of the d’Alembertian. We investigate the equations of motion of the corresponding scalar field, especially in the weak field approximation. We show that this scalar field can exist in a closed and an open FLRW universe with a cosmological constant.