Basic Analytic Number Theory

TL;DR: In this article, the authors considered the problem of the distribution of prime numbers in short intervals and gave an explicit formula for Waring's problem and a new boundary for the Zeros of the Zeta Function.

Abstract: 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 < 4070 and their Smallest Primitive Roots.