Basic Analytic Number Theory - Couverture rigide

Karatsuba, A.A.

Couverture rigide

ISBN 10 : 3540533451 ISBN 13 : 9783540533450

Editeur : Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 1993

Synopsis

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

Détails de l'édition

• Éditeur: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Date d'édition: 1993
• Langue: anglais
• ISBN 10: 3540533451
• ISBN 13: 9783540533450
• Reliure: Relié
• Nombre de pages: 240
• Coordonnées du fabricant: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3,69115 Heidelberg, Germany; Heidelberg; 69115; Allemagne