The Borel Transform and Linear Nonlocal Equations: Applications to Zeta-Nonlocal Field Models

We define rigorously operators of the form f(∂t), in which f is an analytic function on a simply connected domain. Our formalism is based on the Borel transform on entire functions of exponential type. We study existence and regularity of real-valued solutions for the nonlocal in time equation f(∂t)ϕ = J(t), t ∈ R, and we find its more general solution as a restriction to R of an entire function of exponential type. As an important special case, we solve explicitly the linear nonlocal zeta field equation ζ(∂2t + h)ϕ = J(t), in which h is a real parameter, ζ is the Riemann zeta function, and J is an entire function of exponential type. We also analyse the case in which J is a more general analytic function (subject to some weak technical assumptions). This latter case turns out to be rather delicate: we need to re-interpret the symbol ζ(∂2t + h). We prove that in this case the zeta-nonlocal equation above admits an analytic solution on a simply connected domain determined by J. The linear zeta field equation is a linear version of a field model depending on the Riemann zeta function arising in p-adic string theory [B. Dragovich, Zeta-nonlocal scalar fields, Theoret. Math. Phys., 157 (2008), 1671–1677]. © 2022, International Press, Inc.. All rights reserved.