CONTENTS
Translatoes Preface = ⅷ
Introductibn to theEnAlish Editloh = ⅸ
Introduction to ihe'geond Russian 1dltion = ⅹ
Notation = xii
Chapter Ⅰ. Integer Points = 1
1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results = 1
2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums = 6
3. Theorems on Trigonometric Sums = 10
4. Integer Points in a Circle and Under a Hyperbola = 21
Exercises = 25
Chapter Ⅱ. Entire Functions of Finite Order = 27
1. Infinite Products. Weierstrass's Formula = 27
2. Entire Functions of Finite Order = 32
Exercises = 38
Chapter Ⅲ. The Euler Gamma Function = 41
1. Definition and Simplest Properties = 41
2. Stirling's Formula = 44
3. The Euler Beta Function and Dirichlet's Integral = 45
Exercises = 48
Chapter Ⅳ. The Riemann Zeta Function = 51
1. Definition and Simplest Properties = 51
2. Simplest Theorems on the Zeros = 56
3. Approximation by a Finite Sum = 61
Exercises = 62
Chapter Ⅴ. The Connection Between the Sum of the Coefficients of a Dirichiet Series and the Function Defined by this Series = 64
1. A General Theorem = 64
2. The Prime Number Theorem = 66
3. Representation of the Chebyshev Functions as Sums Over the Zeros of the Zeta Function = 69
Exercises = 70
Chapter Ⅵ. The Method of I.M. Vinogradov in the Theory of the Zeta Function = 73
1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum = 73
2. Estimate of a Zeta Sum = 82
3. Estimate for the Zeta Function Close to the Line 1 = 86
4. A Function-Theoretic Lemma = 87
5. A New Boundary for the Zeros of the Zeta Function = 88
6. A New Remainder Term in the Prime, Number Theorem = 90
Exercises = 91
Chapter Ⅶ. The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals = 94
1. The Simplest Density Theorem = 94
2. Prime Numbers in Short Intervals = 98
Exercises = 100
Chapter Ⅷ. Dirichlet L-Functions = 102
1. Characters and their Properties = 102
2. Definition of L-Functions and their Simplest Properties = 110
3. The Functional Equation = 113
4. Non-trivial Zeros: Expansion of the Logarithmic Derivative as a Series in the Zero = 116
5. Simplest Theorems on the Zeros = 117
Exercises = 119
Chapter Ⅸ. Prime Numbers in Arithmetic Progressions = 122
1. An Explicit Formula = 122
2. Theorems on the Boundary of the Zeros = 124
3. The Prime Number Theorem for Arithmetic Progressions = 135
Exercises = 138
Chapter Ⅹ. The Goldbach Conjecture = 141
1. Auxiliary Statements = 141
2. The Circle Method for Goldbach's Problem = 142
3. Linear Trigonometric Sums with Prime Numbers = 149
4. An Effective Theorem = 153
Exercises = 158
Chapter XI. Waring's Problem = 160
1. The Circle Method for Waring's Problem = 160
2. An Estimate for Weyl Sums and the Asymptotic Formula for Waring's Problem = 171
3. An Estimate for G(n) = 174
Exercises = 177
Hints for the Solution of the Exercises = 181
Table of Prime Numbers 4070 and their Smallest Primitive Roots = 217
Bibliography = 219
Subject Index = 221
