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p‐Adic strings are important objects of string theory, as well as of p‐adic mathematical physics and nonlocal cosmology. By a concept of adelic string one can unify and simultaneously study various aspects of ordinary and p‐adic strings. By this way, one can consider adelic strings as a very useful instrument in the further investigation of modern string theory. It is remarkable that for some scalar p‐adic strings exist effective Lagrangians, which are based on real instead of p‐adic numbers and describe not only four‐point scattering amplitudes but also all higher ones at the tree level. In this work, starting from p‐adic Lagrangians, we consider some approaches to construction of effective field Lagrangians for p‐adic sector of adelic strings. It yields Lagrangians for nonlinear and nonlocal scalar field theory, where spacetime nonlocality is determined by an infinite number of derivatives contained in the operator‐valued Riemann zeta function. Owing to the Riemann zeta function in the dynamics of these scalar field theories, obtained Lagrangians are also interesting in themselves.

Towards effective Lagrangians for adelic strings