Author(s): Heirnut Maier, Wolfgang P. Schleich
Title page
Preface
1. Where Are the Primes?
1.1. Ante
l.2. Sieve of Eratosthenes
l.3. Prime Function pi
2. How to Count Primes
2.l. Sieve of Eratosthenes as a Set Problem
2.2. Number of Elements in M₀
2.3. A Compact Formula for π(x)-π(√x)
2.4. A Second Glimpse of the Prime Number Theorem
3. Möbius Function μ
3.l. Multiplicative Arithmetic Functions
3.2. Convolution of Two Arithmetic Functions
3.3. Möbius Inversion Formula
3.4. Generating Functions and Dirichlet Series
3.5. Riemann ζ-Function
4. Analytical Properties of ζ
4.l. Extension of ζ to Whole Complex Plane
4.2. Zeros of ζ
4.3. Mellin Transform
5. Prime Number Theorem
5.l. New Representation of Prime Function
5.2. New Representation of Chebyshev function
5.3. Explicit Asymptotic Expression for π
6. Gauss Sums
6.l. Theta Functions in Various Forms
6.2. Emergence of Gauss Sums
7. In a Nutshell
A. Riemann's Original Paper
B. Inclusion-Exclusion Principle
B.l. Two Different but Equivalent Forms
B.2. Sieve of Brun
C. Improved Mertens' formula
D. General Convolutions
D.1. Convolution for Addition
D.2. Continuous Variables
D.3. Multiplicative Convolution
E. Functional Equations
E.1. Poisson Summation Formula
E.2. Functional Equation for θ
F. Evaluation of Gauss Sums
F.l. Gauss Sum G(l,q)
F.2. Gauss Sum G(a,q)
F.3. Congruences and Quadratic Residues
G. Further Reading
H. Solutions to the Problems
