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Limit Theorems for the Riemann Zeta-Function

  • Book
  • © 1996

Overview

Authors:
  1. Antanas Laurinčikas

    1. Department of Mathematics, Vilnius University, Vilnius, Lithuania

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Part of the book series: Mathematics and Its Applications (MAIA, volume 352)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Elements of the Probability Theory

    • Antanas Laurinčikas
    Pages 1-25

  3. Dirichlet Series and Dirichlet Polynomials

    • Antanas Laurinčikas
    Pages 26-86

  4. Limit Theorems for the Modulus of the Riemann Zeta-Function

    • Antanas Laurinčikas
    Pages 87-148

  5. Limit Theorems for the Riemann Zeta-Function on the Complex Plane

    • Antanas Laurinčikas
    Pages 149-178

  6. Limit Theorems for the Riemann Zeta-Function in the Space of Analytic Functions

    • Antanas Laurinčikas
    Pages 179-202

  7. Universality Theorem for the Riemann Zeta-Function

    • Antanas Laurinčikas
    Pages 203-236

  8. Limit Theorem for the Riemann Zeta-Function in the Space of Continuous Functions

    • Antanas Laurinčikas
    Pages 237-250

  9. Limit Theorems for Dirichlet L-Functions

    • Antanas Laurinčikas
    Pages 251-275

  10. Limit Theorem for the Dirichlet Series with Multiplicative Coefficients

    • Antanas Laurinčikas
    Pages 276-285

  11. Back Matter

    Pages 286-305
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Keywords

About this book

The subject of this book is probabilistic number theory. In a wide sense probabilistic number theory is part of the analytic number theory, where the methods and ideas of probability theory are used to study the distribution of values of arithmetic objects. This is usually complicated, as it is difficult to say anything about their concrete values. This is why the following problem is usually investigated: given some set, how often do values of an arithmetic object get into this set? It turns out that this frequency follows strict mathematical laws. Here we discover an analogy with quantum mechanics where it is impossible to describe the chaotic behaviour of one particle, but that large numbers of particles obey statistical laws. The objects of investigation of this book are Dirichlet series, and, as the title shows, the main attention is devoted to the Riemann zeta-function. In studying the distribution of values of Dirichlet series the weak convergence of probability measures on different spaces (one of the principle asymptotic probability theory methods) is used. The application of this method was launched by H. Bohr in the third decade of this century and it was implemented in his works together with B. Jessen. Further development of this idea was made in the papers of B. Jessen and A. Wintner, V. Borchsenius and B.

Authors and Affiliations

  • Department of Mathematics, Vilnius University, Vilnius, Lithuania

    Antanas Laurinčikas

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