Basic Analytic Number Theory

1. Front Matter

   Pages i-xiii

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2. Integer Points

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 1-26

3. Entire Functions of Finite Order

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 27-40

4. The Euler Gamma Function

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 41-50

5. The Riemann Zeta Function

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 51-63

6. The Connection Between the Sum of the Coefficients of a Dirichlet Series and the Function Defined by this Series

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 64-72

7. The Method of I.M. Vinogradov in the Theory of the Zeta Function

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 73-93

8. The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 94-101

9. Dirichlet L-Functions

   • Anatolij A. Karatsuba, Melvyn B. Nathanson

   Pages 102-121

10. Prime Numbers in Arithmetic Progressions

    • Anatolij A. Karatsuba, Melvyn B. Nathanson

    Pages 122-140

11. The Goldbach Conjecture

    • Anatolij A. Karatsuba, Melvyn B. Nathanson

    Pages 141-159

12. Waring’s Problem

    • Anatolij A. Karatsuba, Melvyn B. Nathanson

    Pages 160-180

13. Back Matter

    Pages 181-222

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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.

• Analytic Number Theory
• Analytische Zahlentheorie
• Distribution of prime numbers
• Goldbach's Problem
• Prime
• Prime number
• Waring's problem
• number theory
• the circle method

• Steklov Mathematical Institute, Moscow, Russia

  Anatolij A. Karatsuba

• Institute for Advanced Study, School of Mathematics, Princeton, USA

  Melvyn B. Nathanson

• Department of Mathematics, Lehman College (CUNY), Bronx, USA

  Melvyn B. Nathanson

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