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AuthorKaratsuba, Anatolij A. author
TitleBasic Analytic Number Theory [electronic resource] / by Anatolij A. Karatsuba, Melvyn B. Nathanson
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1993
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Descript XIII, 222 p. online resource


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 transยญ 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


I. Integer Points -- ยง1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results -- ยง2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums -- ยง3. Theorems on Trigonometric Sums -- ยง4. Integer Points in a Circle and Under a Hyperbola -- Exercises -- II. Entire Functions of Finite Order -- ยง1. Infinite Products. Weierstrassโ{128}{153}s Formula -- ยง2. Entire Functions of Finite Order -- Exercises -- III. The Euler Gamma Function -- ยง1. Definition and Simplest Properties -- ยง2. Stirlingโ{128}{153}s Formula -- ยง3. The Euler Beta Function and Dirichletโ{128}{153}s Integral -- Exercises -- IV. The Riemann Zeta Function -- ยง1. Definition and Simplest Properties -- ยง2. Simplest Theorems on the Zeros -- ยง3. 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 -- ยง1. A General Theorem -- ยง2. The Prime Number Theorem -- ยง3. 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 -- ยง1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum -- ยง2. Estimate of a Zeta Sum -- ยง3. Estimate for the Zeta Function Close to the Line ? = 1 -- ยง4. A Function-Theoretic Lemma -- ยง5. A New Boundary for the Zeros of the Zeta Function -- ยง6. 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 -- ยง1. The Simplest Density Theorem -- ยง2. Prime Numbers in Short Intervals -- Exercises -- VIII. Dirichlet L-Functions -- ยง1. Characters and their Properties -- ยง2. Definition of L-Functions and their Simplest Properties -- ยง3. The Functional Equation -- ยง4. Non-trivial Zeros; Expansion of the Logarithmic Derivative as a Series in the Zeros -- ยง5. Simplest Theorems on the Zeros -- Exercises -- IX. Prime Numbers in Arithmetic Progressions -- ยง1. An Explicit Formula -- ยง2. Theorems on the Boundary of the Zeros -- ยง3. The Prime Number Theorem for Arithmetic Progressions -- Exercises -- X. The Goldbach Conjecture -- ยง1. Auxiliary Statements -- ยง2. The Circle Method for Goldbachโ{128}{153}s Problem -- ยง3. Linear Trigonometric Sums with Prime Numbers -- ยง4. An Effective Theorem -- Exercises -- XI. Waringโ{128}{153}s Problem -- ยง1. The Circle Method for Waringโ{128}{153}s Problem -- ยง2. An Estimate for Weyl Sums and the Asymptotic Formula for Waringโ{128}{153}s Problem -- ยง3. An Estimate for G(n) -- Exercises -- Hints for the Solution of the Exercises -- Table of Prime Numbers < 4070 and their Smallest Primitive Roots

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