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. ...
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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
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