Titlebook: Limit Theorems for the Riemann Zeta-Function; Antanas Laurin?ikas Book 1996 Springer Science+Business Media Dordrecht 1996 Rang.number the

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Mathematics and Its Applicationshttp://image.papertrans.cn/l/image/586171.jpg

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Limit Theorems for the Riemann Zeta-Function on the Complex Plane,Since the function .(.) is complex-valued the distribution of its values is reflected more adequately by limit theorems on the complex plane C. In this chapter we will obtain such theorems on the half-plane . ≥ 1/2.

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Limit Theorems for the Riemann Zeta-Function in the Space of Analytic Functions,Let .. = {. ∈ .: 1/2 < . < 1} and .. = {. ∈ .: . > 1}. We define the probability measures . on (.(..), .(.(..))), . = 1, 2. The aim of this chapter is to prove that the measures .., converge weakly to some measure as . → ∞. Let . = {. ∈ .: . > 1/2}.

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Universality Theorem for the Riemann Zeta-Function,In this chapter we apply the limit theorem for the Riemann zeta-function in the space .(..) to obtain one of magnificent properties of this function — the universality property. Roughly speaking, this property asserts that any analytic function can be approximated uniformly on compact subsets of .. by translations of ζ(.).

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Limit Theorems for Dirichlet ,-Functions,The properties of Dirichlet .-functions with a fixed modulus are similar to those of the Riemann zeta-function. Therefore all limit theorems proved in the previous chapter can be stated also for Dirichlet .-functions. In this chapter we will give only few limit theorems which describe the asymptotic behaviour of the Dirichlet .-functions.

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978-90-481-4647-5Springer Science+Business Media Dordrecht 1996

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Elements of the Probability Theory,ying of the distribution of values of some functions defined by the Dirichlet series. Most of the material consists of well-known facts, and their proofs can be found in monographs on the theory of probability.