{"product_id":"basic-analytic-number-theory-anatolij-a-karatsuba-9783540533450","title":"Basic Analytic Number Theory","description":"I. Integer Points.-  1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results.-  2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums.-  3. Theorems on Trigonometric Sums.-  4. Integer Points in a Circle and Under a Hyperbola.- Exercises.- II. Entire Functions of Finite Order.-  1. Infinite Products. Weierstrass's Formula.-  2. Entire Functions of Finite Order.- Exercises.- III. The Euler Gamma Function.-  1. Definition and Simplest Properties.-  2. Stirling's Formula.-  3. The Euler Beta Function and Dirichlet's Integral.- Exercises.- IV. The Riemann Zeta Function.-  1. Definition and Simplest Properties.-  2. Simplest Theorems on the Zeros.-  3. 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.-  1. A General Theorem.-  2. The Prime Number Theorem.-  3. 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.-  1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum.-  2. Estimate of a Zeta Sum.-  3. Estimate for the Zeta Function Close to the Line ? = 1.-  4. A Function-Theoretic Lemma.-  5. A New Boundary for the Zeros of the Zeta Function.-  6. 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.-  1. The Simplest Density Theorem.-  2. Prime Numbers in Short Intervals.- Exercises.- VIII. Dirichlet L-Functions.-  1. Characters and their Properties.-  2. Definition of L-Functions and their Simplest Properties.-  3. The Functional Equation.-  4. Non-trivial Zeros; Expansion of the Logarithmic Derivative as a Series in the Zeros.-  5. Simplest Theorems on the Zeros.- Exercises.- IX. Prime Numbers in Arithmetic Progressions.-  1. An Explicit Formula.-  2. Theorems on the Boundary of the Zeros.-  3. The Prime Number Theorem for Arithmetic Progressions.- Exercises.- X. The Goldbach Conjecture.-  1. Auxiliary Statements.-  2. The Circle Method for Goldbach's Problem.-  3. Linear Trigonometric Sums with Prime Numbers.-  4. An Effective Theorem.- Exercises.- XI. Waring's Problem.-  1. The Circle Method for Waring's Problem.-  2. An Estimate for Weyl Sums and the Asymptotic Formula for Waring's Problem.-  3. An Estimate for G(n).- Exercises.- Hints for the Solution of the Exercises.- Table of Prime Numbers\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Anatolij A. Karatsuba\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 3540533451\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9783540533450\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Springer\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 11\/19\/1992\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 240\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Hardcover\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 1.13lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.21h x 6.14w x 0.56d","brand":"Anatolij A. Karatsuba","offers":[{"title":"Hardcover","offer_id":48742748324095,"sku":"9783540533450","price":99.0,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_7b790380-e57a-45da-a7a5-31ea398a1a20.jpg?v=1782447555","url":"https:\/\/www.whiterainbookhouse.com\/products\/basic-analytic-number-theory-anatolij-a-karatsuba-9783540533450","provider":"WR Book House","version":"1.0","type":"link"}