β Scribed by Antanas LaurinΔikas (auth.)
No coin nor oath required. For personal study only.
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.
Front Matter....Pages i-xiii
Elements of the Probability Theory....Pages 1-25
Dirichlet Series and Dirichlet Polynomials....Pages 26-86
Limit Theorems for the Modulus of the Riemann Zeta-Function....Pages 87-148
Limit Theorems for the Riemann Zeta-Function on the Complex Plane....Pages 149-178
Limit Theorems for the Riemann Zeta-Function in the Space of Analytic Functions....Pages 179-202
Universality Theorem for the Riemann Zeta-Function....Pages 203-236
Limit Theorem for the Riemann Zeta-Function in the Space of Continuous Functions....Pages 237-250
Limit Theorems for Dirichlet L -Functions....Pages 251-275
Limit Theorem for the Dirichlet Series with Multiplicative Coefficients....Pages 276-285
Back Matter....Pages 286-305
Number Theory; Probability Theory and Stochastic Processes; Functions of a Complex Variable; Functional Analysis; Measure and Integration
π SIMILAR VOLUMES
This is a text on the mean-value and omega theorems for the Riemann Zeta-function. It includes discussion of some fundamental theorems on Titchmarsh series and applications, and Titchmarsh's Phenomenon.
This book provides both classical and new results in Reimann Zeta-Function theory, one of the most important problems in analytic number theory. These results have application in solving problems in multiplicative number theory, such as power moments, the zero-free region, and the zero density estim
<p>"[β¦] the scope of this well-written book is by no means restricted to the Riemann zeta-function. It spans the range successfully from elementary theory to topics of recent and current research." <em>Mathematical Reviews</em> </p>