Basic analytic number theory.

Includes index.

Includes Bibliographical references (p. 219)

I. Integer Points
ʹ1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results
ʹ2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums
ʹ3. Theorems on Trigonometric Sums
ʹ4. Integer Points in a Circle and Under a Hyperbola Exercises
II. Entire Functions of Finite Order
ʹ1. Infinite Products. Weierstrass's Formula
ʹ2. Entire Functions of Finite Order
III. The Euler Gamma Function
1. Definition and Simplest Properties
ʹ2. Stirling's Formula
ʹ3. The Euler Beta Function and Dirichlet's Integral
IV. The Riemann Zeta Function
ʹ2. Simplest Theorems on the Zeros
ʹ3. Approximation by a Finite Sum
V. The Connection Between the Sum of the Coefficients of a Dirichlet Series and the Function Defined by this Series
ʹ1. A General Theorem
ʹ2. The Prime Number Theorem
ʹ3. Representation of the Chebyshev Functions as Sums Over the Zeros of the Zeta Function
VI. The Method of I.M. Vinogradov in the Theory of the Zeta Function
ʹ1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum
ʹ2. Estimate of a Zeta Sum
ʹ3. Estimate for the Zeta Function Close to the Line? = 1
ʹ4. A Function-Theoretic Lemma
ʹ5. A New Boundary for the Zeros of the Zeta Function
ʹ6. A New Remainder Term in the Prime Number Theorem
VII. The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals
ʹ1. The Simplest Density Theorem
ʹ2. Prime Numbers in Short Intervals
VIII. Dirichlet L-Functions
ʹ1. Characters and their Properties
ʹ2. Definition of L-Functions and their Simplest Properties
ʹ3. The Functional Equation
ʹ4. Non-trivial Zeros; Expansion of the Logarithmic Derivative as a Series in the Zeros
ʹ5. Simplest Theorems on the Zeros IX. Prime Numbers in Arithmetic Progressions
ʹ1. An Explicit Formula
ʹ2. Theorems on the Boundary of the Zeros
ʹ3. The Prime Number Theorem for Arithmetic Progressions X. The Goldbach Conjecture
ʹ1. Auxiliary Statements
ʹ2. The Circle Method for Goldbach's Problem
ʹ3. Linear Trigonometric Sums with Prime Numbers
ʹ4. An Effective Theorem XI. Waring's Problem
ʹ1. The Circle Method for Waring's Problem
ʹ2. An Estimate for Weyl Sums and the Asymptotic Formula for Waring's Problem
ʹ3. An Estimate for G(n).

This English translation of Karatsuba's Basic Analytic Number Theory follows closely the second Russian edition, published in Moscow in 1983. For the English edition, the author has considerably rewritten Chapter I, and has corrected various typographical and other minor errors throughout the the text. August, 1991 Melvyn B. Nathanson Introduction to the English Edition It gives me great pleasure that Springer-Verlag is publishing an English trans­ lation of my book. In the Soviet Union, the primary purpose of this monograph was to introduce mathematicians to the basic results and methods of analytic number theory, but the book has also been increasingly used as a textbook by graduate students in many different fields of mathematics. I hope that the English edition will be used in the same ways. I express my deep gratitude to Professor Melvyn B. Nathanson for his excellent translation and for much assistance in correcting errors in the original text. A.A. Karatsuba Introduction to the Second Russian Edition Number theory is the study of the properties of the integers. Analytic number theory is that part of number theory in which, besides purely number theoretic arguments, the methods of mathematical analysis play an essential role.