Abstract
In the paper, a joint limit theorem in the sense of weak convergence of probability measures on the complex plane for Laplace transforms of the Riemann zetafunction is obtained.
Highlights
Let Z, N, R and C stand for the sets of all integers, positive integers, real and complex numbers, respectively, and let, as usual, ζ(s), s = σ + it, denote the Riemann zetafunction
It is well known that almost periodic functions have limit distributions in the sense of Theorem 1
Almost periodic functions are approximated in some metric by trigonometric polynomials, for Laplace transforms this is not known
Summary
+ε t ≥ t0 > 0, with every ε > 0, the integral for Lk(s) converges absolutely and uniformly on compact subsets of the half – plane D = {s ∈ C : σ > 0}, and defines there an analytic function. Let meas{A} denote the Lebesgue measure of a measurable set A ⊂ R, and let, for T > 0, νT Denote B(S) the class of Borel sets of the space S, and define the probability measure. It is well known that almost periodic functions have limit distributions in the sense of Theorem 1. The majority of functions defined by Dirichlet series have the above property. Almost periodic functions are approximated in some metric by trigonometric polynomials, for Laplace transforms this is not known. We can not apply the almost periodicity property for Laplace transforms
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