Complex Analysis in Number Theory
| Date: 16 April 2011, 00:28
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This book examines the application of complex analysis methods to the theory of prime numbers. In an easy to understand manner, a connection is established between arithmetic problems and those of zero distribution for special functions. Main achievements in this field of mathematics are described. Indicated is a connection between the famous Riemann zeta-function and the structure of the universe, information theory, and quantum mechanics. The theory of Riemann zeta-function and, specifically, distribution of its zeros are presented in a concise and comprehensive way. The full proofs of some modern theorems are given. Significant methods of the analysis are also demonstrated as applied to fundamental problems of number theory.
The application of the methods of mathematical analysis and, in
particular, of the theory of functions of a complex variable to the
solution of problems of the number theory contributed to a considerable
progress in this branch of mathematics. Here is what I.M. Vinogradov
said in this connection in his review entitled "On the Problems
of the Analytic Number Theory" published in the proceedings of the
November jubilee session of the USSR Academy of Sciences dedicated
to the 15th anniversary of the USSR: "Analysis makes it possible to
extend considerably the range of problems of the number theory and
provides for a more rapid development of this science. I also want
to point out one more useful feature of the analytic method in the
number theory. While solving new difficult problems, analysis itself
develops and gets more perfect. Dirichlet's series and the theory of
the ((s) function can serve as examples as well as some properties of
Bessel's functions, a number of remarkable theorems relating to the
theory of functions of a complex variable (for instance, the theorems
of Lindelof, Phragmen, Mellin), discontinuous sums and integrals etc.
Thus, the application of the analytic method to the number theory
enriches this science with new valuable achievements and, at the same
time, develops and perfects the analysis itself."
This mono$aph primarily deals with the application of analysis
to the problems of the theory of prime numbers. At the same time,
it presents the results that appeared in the framework of the number
theory but actually refer to analysis. The monograph does not cover
the whole wealth of material connected with the indicated problems
of the number theory.
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