Basic Analytic Number Theory

A.A. Karatsuba

ISBN 10: 3540533451 / ISBN 13: 9783540533450

Verlag: Springer-Verlag Berlin and Heidelberg GmbH and Co. KG, DE, 1993

Beschreibung

This work provides an introduction to four central problems in analytic number theory. These are (1) the problems of estimating the number of integer points in planar domains, (2) the problem of the distribution of prime numbers in the sequence of all natural numbers and in arithmetic progressions, (3) Goldbach's problems on sums of primes, and (4) Waring's problem on sums of k-th powers. The following fundamental methods of analytic number theory are used to solve these problems: complex integration, I.M. Vinogradov's method of trigonometric sums, and the circle method of G.H. Hardy, J.E. Littlewood, and S. Ramanujan. There are numerous exercises at the end of each chapter. These exercises either refine the theorems proved in the text, or lead to new ideas in number theory. The author also includes a section of hints for the solution of the exercises.

Bestandsnummer des Verkäufers LU-9783540533450

Bibliografische Details

• Titel: Basic Analytic Number Theory
• Verlag: Springer-Verlag Berlin and Heidelberg GmbH and Co. KG, DE
• Erscheinungsjahr: 1993
• Sprache: Englisch
• ISBN-10: 3540533451
• ISBN-13: 9783540533450
• Einband: Hardback
• Auflage: Illustrated.
• Zustand: New

Über diesen Titel

Inhaltsangabe

I. Integer Points.- §1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results.- §2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums.- §3. Theorems on Trigonometric Sums.- §4. Integer Points in a Circle and Under a Hyperbola.- Exercises.- II. Entire Functions of Finite Order.- §1. Infinite Products. Weierstrass's Formula.- §2. Entire Functions of Finite Order.- Exercises.- III. The Euler Gamma Function.- §1. Definition and Simplest Properties.- §2. Stirling's Formula.- §3. The Euler Beta Function and Dirichlet's Integral.- Exercises.- IV. The Riemann Zeta Function.- §1. Definition and Simplest Properties.- §2. Simplest Theorems on the Zeros.- §3. Approximation by a Finite Sum.- Exercises.- V. The Connection Between the Sum of the Coefficients of a Dirichlet Series and the Function Defined by this Series.- §1. A General Theorem.- §2. The Prime Number Theorem.- §3. Representation of the Chebyshev Functions as Sums Over the Zeros of the Zeta Function.- Exercises.- VI. The Method of I.M. Vinogradov in the Theory of the Zeta Function.- §1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum.- §2. Estimate of a Zeta Sum.- §3. Estimate for the Zeta Function Close to the Line ? = 1.- §4. A Function-Theoretic Lemma.- §5. A New Boundary for the Zeros of the Zeta Function.- §6. A New Remainder Term in the Prime Number Theorem.- Exercises.- VII. The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals.- §1. The Simplest Density Theorem.- §2. Prime Numbers in Short Intervals.- Exercises.- VIII. Dirichlet L-Functions.- §1. Characters and their Properties.- §2. Definition of L-Functions and their Simplest Properties.- §3. The Functional Equation.- §4. Non-trivial Zeros; Expansion of the Logarithmic Derivative as a Series in the Zeros.- §5. Simplest Theorems on the Zeros.- Exercises.- IX. Prime Numbers in Arithmetic Progressions.- §1. An Explicit Formula.- §2. Theorems on the Boundary of the Zeros.- §3. The Prime Number Theorem for Arithmetic Progressions.- Exercises.- X. The Goldbach Conjecture.- §1. Auxiliary Statements.- §2. The Circle Method for Goldbach's Problem.- §3. Linear Trigonometric Sums with Prime Numbers.- §4. An Effective Theorem.- Exercises.- XI. Waring's Problem.- §1. The Circle Method for Waring's Problem.- §2. An Estimate for Weyl Sums and the Asymptotic Formula for Waring's Problem.- §3. An Estimate for G(n).- Exercises.- Hints for the Solution of the Exercises.- Table of Prime Numbers < 4070 and their Smallest Primitive Roots.

Reseña del editor

This work provides an introduction to four central problems in analytic number theory. These are (1) the problems of estimating the number of integer points in planar domains, (2) the problem of the distribution of prime numbers in the sequence of all natural numbers and in arithmetic progressions, (3) Goldbach's problems on sums of primes, and (4) Waring's problem on sums of k-th powers. The following fundamental methods of analytic number theory are used to solve these problems: complex integration, I.M. Vinogradov's method of trigonometric sums, and the circle method of G.H. Hardy, J.E. Littlewood, and S. Ramanujan. There are numerous exercises at the end of each chapter. These exercises either refine the theorems proved in the text, or lead to new ideas in number theory. The author also includes a section of hints for the solution of the exercises.