A GENERALIZATION OF FUNCTIONAL LIMIT THEOREMS ON THE RIEMANN ZETA PROCESS

zeta(.) being the Riemann zeta function, zeta(sigma)(t) := zeta(sigma+it)/zeta(sigma) is, for sigma > 1, a characteristic function of some infinitely divisible distribution mu(sigma). A process with time parameter sigma having mu(sigma) as its marginal at time sigma is called a Riemann zeta process. Ehm [2] has found a functional limit theorem on this process being a backwards Levy process. In this paper, we replace zeta(.) with a Dirichlet series eta(.; a) generated by a nonnegative, completely multiplicative arithmetical function a(.) satisfying (3), (4) and (5) below, and derive the same type of functional limit theorem as Ehm on the process corresponding to eta(.; a) and being a backwards Levy process.