A Riemann zeta stochastic process

and thus be represented (for σ > 1 ) as a product of terms of the form exp(a(eibt − 1)), each of which is the characteristic function of a Poisson random variable with intensity a and values in the lattice kb, k = 0, 1, 2, . . . . Cf. Gnedenko and Kolmogorov [6, p. 75]. Faced with a family of “zeta distributions” indexed by parameter σ > 1 , one is led to ask for joint distributions, i.e., for a stochastic process with time parameter σ having these distributions as its marginals. Such a “zeta process” was constructed by Alexander, Baclawski and Rota [1] for discrete time indices σ = 2, 3, . . . . Our object here is to propose an elementary construction of a continuous time zeta process. Of particular interest is, then, its limiting behavior as σ ↓ 1 . Instead of expanding log(1 − p−σ−it) in a Taylor series as in (1), we stay with the initial product representation and note that each factor, (1−p−σ)/(1−p−σ−it) , represents the characteristic function of a random variable of the form −Yp(p) log p , where Y (u) denotes a geometrically distributed random variable with parameter u ∈ (0, 1) , P [Y (u) = n] = (1− u)un (n = 0, 1, 2, . . .) . Therefore, up to a convenient change of sign, a random variable Z(σ) with characteristic function ζσ(t) can be represented in the form