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1992 | Buch

Complete Exponential Sums Kapitel lesen Erstes Kapitel lesen

Exponential Sums and their Applications

verfasst von: N. M. Korobov

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

The method of exponential sums is a general method enabling the solution of a wide range of problems in the theory of numbers and its applications. This volume presents an exposition of the fundamentals of the theory with the help of examples which show how exponential sums arise and how they are applied in problems of number theory and its applications.
The material is divided into three chapters which embrace the classical results of Gauss, and the methods of Weyl, Mordell and Vinogradov; the traditional applications of exponential sums to the distribution of fractional parts, the estimation of the Riemann zeta function; and the theory of congruences and Diophantine equations. Some new applications of exponential sums are also included.
It is assumed that the reader has a knowledge of the fundamentals of mathematical analysis and of elementary number theory.

Inhaltsverzeichnis

Frontmatter
Chapter I. Complete Exponential Sums
Abstract
The simplest example of Weyl’s sums is the sum of the first degree
$$ S(P) = \sum\limits_{x = Q + 1}^{Q + P} {{e^{2\pi i\alpha x}}.} $$
.
N. M. Korobov
Chapter II. Weyl’s Sums
Abstract
In Chapter I, the Weyl sums of the first degree were considered and it was shown that the estimate
$$ \left| {\sum\limits_{x = 1}^P {{e^{2\pi i\alpha x}}} } \right| \leqslant \min \left( {P,\frac{1}{{2\left\| \alpha \right\|}}} \right) $$
holds for them. The basic idea of Weyl’s method consists in reducing the estimation of a sum of an arbitrary degree n ⩾ 2
$$ S\left( P \right) = \sum\limits_{x = 1}^P {{e^{2\pi i\left( {{\alpha _1}x + \ldots + {\alpha _n}{x^n}} \right)}}} $$
to the estimation of a sum of degree n − 1 and, ultimately, to the use of the estimate (140). We have already met the reduction of the degree of an exponential sum in proving the theorem on the modulus of the Gauss sum. In the Gauss theorem, the square of the modulus of the exponential sum of the second degree was transformed with the help of linear change of variable in summation into a double sum, in which one of summations was reduced to the evaluation of a sum of the first degree. Similar but technically more complicated considerations are used for the reduction of the degree of sums in the Weyl method as well.
N. M. Korobov
Chapter III. Fractional Parts Distribution, Normal Numbers, and Quadrature Formulas
Abstract
The notion of uniform distribution in a general form was introduced by H. Weyl [49]. He obtained also fundamental results concerning functions, whose fractional parts are uniformly distributed.
N. M. Korobov
Backmatter
Metadaten
Titel
Exponential Sums and their Applications
verfasst von
N. M. Korobov
Copyright-Jahr
1992
Verlag
Springer Netherlands
Electronic ISBN
978-94-015-8032-8
Print ISBN
978-90-481-4137-1
DOI
https://doi.org/10.1007/978-94-015-8032-8