Abstract
For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic
strings in curved target spacetime, and show that the vacuum Einstein equations
of the target are a consequence of worldsheet scaling symmetry of the quantum
$p$-adic strings, similar to the ordinary bosonic strings case. It turns out
that certain $p$-adic automorphic forms are the plane wave modes of the bosonic
fields on $p$-adic strings, and that the regularized normalization of these
modes on the $p$-adic worldsheet presents peculiar features which reduce part
of the computations to familiar setups in quantum field theory, while also
exhibiting some new features that make loop diagrams much simpler. Assuming a
certain product relation, we also observe that the adelic spectrum of the
bosonic string corresponds to the nontrivial zeros of the Riemann Zeta
function.