Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

Abstract

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the form$f\left({\partial }_t\right)\varphi =J\left(t\right),t\ge 0,$ where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the function$\varphi \left(t\right)=\sum _{n\ge 1}\frac{\mu \left(n\right)}{n^h}J(t-\ln (n\left)\right),$ in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equation$\zeta ({\partial }_t+h)\varphi =J\left(t\right),t\ge 0,$ in which ζ is the Riemann zeta function and $h>1$. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space $L^2$ and the Hardy space $H^2$.