Analytic Number Theory

The following sections are included:

• Preface

• Contents

The following sections are included:

• Three problems

• Asymmetric distribution of quadratic residues

• The prime number theorem

• Density of squarefree integers

• The Riemann zeta function

• Notes

The following sections are included:

• Arithmetic functions and convolution

• Inverses

• Convergence

• Exponential mapping
  • The 1 function as an exponential
  • Powers and roots

• Multiplicative functions

• Notes

The following sections are included:

• Generalities

• Estimate of Q(x) - 6x/π²

• Riemann-Stieltjes integrals

• Riemann-Stieltjes integrators
  • Convolution of integrators
  • Generalization of results on arithmetic functions

• Stability

• Dirichlet's hyperbola method

• Notes

The following sections are included:

• General remarks

• The Chebyshev ψ function

• Mertens' estimates

• Convergent sums over primes

• A lower estimate for Euler's φ function

• Notes

The following sections are included:

• Selberg's formula
  • Features of Selberg's formula

• Transformation of Selberg's formula
  • Calculus for R

• Deduction of the P.N.T.

• Propositions "equivalent" to the P.N.T.

• Some consequences of the P.N.T.

• Notes

The following sections are included:

• The use of transforms

• Euler products

• Convergence
  • Abscissa of convergence
  • Abscissa of absolute convergence

• Uniform convergence

• Analyticity
  • Analytic continuation
  • Continuation of zeta
  • Example of analyticity on σ = σ_c

• Uniqueness
  • Identifying an arithmetic function

• Operational calculus

• Landau's oscillation theorem

• Notes

The following sections are included:

• The use of inversion formulas

• The Wiener-Ikehara theorem
  • Example. Counting product representations
  • An O-estimate

• A Wiener-Ikehara proof of the P.N.T.

• A generalization of the Wiener-Ikehara theorem

• The Perron formula

• Proof of the Perron formula

• Contour deformation in the Perron formula
  • The Fourier series of the sawtooth function
  • Bounded and uniform convergence

• A "smoothed" Perron formula

• Example. Estimation of ∑T(1₂ * 1₃)

• Notes

The following sections are included:

• The functional equation
  • Justification of the interchange of ∑ and ∫
  • Symmetric form of the functional equation

• O-estimates for zeta

• Zeros of zeta

• A zerofree region for zeta

• An estimate of ζ'/ζ

• Estimation of ψ

• The P.N.T. with a remainder term

• Estimation of M

• The density of zeros in the critical strip

• An explicit formula for ψ₁

• Notes

The following sections are included:

• Residue characters

• Group structure of the coprime residue classes

• Existence of enough characters

• L functions

• Proof of Dirichlet's theorem

• P.N.T. for arithmetic progressions

• Notes

The following sections are included:

• Integers generated by primes in residue classes

• Sums of squares

• A measure of nonprincipality

• Quadratic excess

• Evaluation of Gaussian sums

• Notes

The following sections are included:

• Introduction

• Approximate periodicity

• The use of Landau's oscillation theorem

• A quantitative estimate

• The use of many singularities

• Applications

• Sign changes of π(x) - li x

• The size of
  • Numerical calculations

• The error term in the divisor problem

• Notes

The following sections are included:

• Introduction

• The sieve of Eratosthenes and Legendre

• Sieve setup

• The Brun-Hooley sieve

• The large sieve

• An extremal majorant

• Proof of Theorem 12.9

• Notes

The following sections are included:

• A Brun-Hooley estimate of twin primes

• The Brun-Titchmarsh inequality

• Primes represented by polynomials

• A uniform two residue sieve estimate

• Twin primes and Goldbach's problem

• A heuristic formula for twin primes

• Notes

The following sections are included:

• Appendix A Results from Analysis and Algebra
  • Properties of real functions
    • Decomposition
    • Riemann-Stieltjes integrals
    • Integrators
  • The Euler gamma function
  • Poisson summation formula
  • Basis theorem for finite abelian groups

• Bibliography

• Index of Names and Topics

• Index of Symbols