Author | Karatsuba, Anatolij A. author |
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Title | Basic Analytic Number Theory [electronic resource] / by Anatolij A. Karatsuba, Melvyn B. Nathanson |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1993 |
Connect to | http://dx.doi.org/10.1007/978-3-642-58018-5 |
Descript | XIII, 222 p. online resource |
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. Weierstrassโs Formula -- ยง2. Entire Functions of Finite Order -- Exercises -- III. The Euler Gamma Function -- ยง1. Definition and Simplest Properties -- ยง2. Stirlingโs Formula -- ยง3. The Euler Beta Function and Dirichletโs 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 Goldbachโs Problem -- ยง3. Linear Trigonometric Sums with Prime Numbers -- ยง4. An Effective Theorem -- Exercises -- XI. Waringโs Problem -- ยง1. The Circle Method for Waringโs Problem -- ยง2. An Estimate for Weyl Sums and the Asymptotic Formula for Waringโs 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