Steps into Analytic Number Theory A Problem-Based Introduction

Description

Preface Notation Contents Step #1 Hello to Big-Oh Asymptotic Analysis Ingenuity Step #2 Asymptotic Analysis Infinitely Many Primes Combinatorial Methods Ingenuity Step #3 Asymptotic Analysis Combinatorial Methods Arithmetic Functions and the Anatomy of Integers Ingenuity Step #4 Variations on a Theme of Euler Arithmetic Functions and the Anatomy of Integers Computing with Roots of Unity Dirichlet Series Mathematical Masterpieces: The Identity as Art Form Step #5 Distribution of Squares mod p Variations on a Theme of Euler Arithmetic Functions and the Anatomy of Integers Dirichlet Series Mathematical Masterpieces: The Identity as Art Form Step #6 Distribution of Squares mod p Combinatorial Methods Mathematical Masterpieces: The Identity as Art Form Ingenuity Step #7 Distribution of Prime Numbers Combinatorial Methods Mathematical Masterpieces: The Identity as Art Form Ingenuity Step #8 Distribution of Squares mod p Distribution of Prime Numbers Variations on a Theme of Euler Ingenuity Step #9 Arithmetic Functions and the Anatomy of Integers Combinatorial Methods Distribution of Squares mod p Mathematical Masterpieces: The Identity as Art Form Ingenuity Step #10 Variations on a Theme of Euler Arithmetic Functions and the Anatomy of Integers Order of 2 mod Mathematical Masterpieces: The Identity as Art Form Ingenuity Step #11 Distribution of Squares mod p Arithmetic Functions and the Anatomy of Integers Order of 2 mod Step #12 Arithmetic Functions and the Anatomy of Integers Distribution of Prime Numbers Ingenuity Special Step A: Dirichlet's Theorem for m=8 Characters of U8 Dirichlet L-Functions for m=8 Reduction to the Nonvanishing of L(1,χ) for χ=χ0 Nonvanishing of L(1,χ) for χ=χ0 Techniques of Generalization Special Step B: Dirichlet's Theorem for m= (Odd Prime) Characters of U The L-Functions L(s,χ) Reduction to the Nonvanishing of L(1,χ) for χ=χ0 Nonvanishing of L(1,χ) for χ=χ0 Ingenuity Special Step C: Dirichlet's Theorem in the General Case Character Table Magic Nonvanishing of L(1,χ) for χ=χ0 Ingenuity Solutions to Step #1 Reference Solutions to Step #2 References Solutions to Step #3 Solutions to Step #4 Reference Solutions to Step #5 Reference Solutions to Step #6 References Solutions to Step #7 Solutions to Step #8 References Solutions to Step #9 Solutions to Step #10 References Solutions to Step #11 References Solutions to Step #12 Reference Solutions to Special Step A References Solutions to Special Step B References Solutions to Special Step C References Epilogue Reference Suggestions for Further Reading