302 Index

character of 2, 47, 68, 83

character of 3, 69, 83

Cunningham-Gosset criterion, 75

cyclotomic numbers, 61

determination when e = 3, 65

cyclotomic polynomials

definition, 24

form of prime divisors, 25

have integer coeﬃcients, 24

irreducibility, 80

cyclotomy, 46

deficient numbers, 248

density, asymptotic, xiii

Dirichlet L-series

nonvanishing at s = 1 for complex χ, 128

nonvanishing at s = 1 for real χ, 132

Dirichlet characters, 126

modulo 4, 120

orthogonality relations for, 127

Dirichlet series, 5, 221

Dirichlet’s theorem, 23, 119

for progressions modulo 4, 120

distribution function, 268

Erd˝ os–Wintner theorem, 268

for σ(n)/n, 252, 259, 273, 274

divisor function, 114

dual group, 125

Elliott–Halberstam conjecture, 109

Erd˝ os–Kac theorem, 112

Erd˝ os–Straus conjecture, 174, 207

Erd˝ os–Wintner theorem, 268

Euler factorization, 5

Euler’s prime-producing polynomial, 14

Extended Riemann Hypothesis, 143

Farey fraction, 145

Fermat number, 29

Fibonacci number, 203

Gauss sum, 81, 146

Gauss–Wantzel theorem, see also

constructibility of regular n-gon

(Gauss–Wantzel characterization)

Gaussian period, 54

period polynomial, 57

form of prime divisors (Kummer’s

criterion), 59

form when e = 2, 61

form when e = 3, 64

has integer coeﬃcients and is

irreducible, 58

reduced period polynomial, 57

form when e = 2, 61

form when e = 3, 68

Gelfond–Schneider transcendence theorem,

33

Goldbach conjecture

lower bound on the number of

representations as a sum of almost

primes, 196

quantitative form, 103, 209

upper bound on the number of

representations, 185

Hasse–Minkowski theorem, 140

Hilbert–Dress identities, 152

Hilbert–Waring theorem, 151

Hypothesis H, 27, 28

quantitative form, 103

implied constant, xiii

Jacobson radical, 37

Legendre’s theorem on diagonal ternary

quadratic forms, 135

Linnik’s theorem on the least prime in a

progression, 143

little-oh notation, xii, xiii

logarithmic integral, 86

M¨ obius inversion, 218

Mann’s theorem, 198

Mann–Shanks primality criterion, 43

Matijasevich–Putnam theorem, 32

Mersenne number, 29

Mersenne prime, 29

Mertens’ theorems, 95

Mertens’ first theorem, 96

Mertens’ second theorem, 97

second theorem for arithmetic

progressions, 141

second theorem for polynomials, 116

multiplication table, 112

multiply perfect number, 272

normal number, 34

normal number of prime factors

of p − 1, 207

of a natural number, 111

O and o notation, xiii

P´ olya–Vinogradov inequality, 146

perfect numbers, 174, 248

conjectured number up to x, 249

Dickson’s theorem, 250

generalization by Kanold, 267

proof of, 253

Euclid–Euler classification of even

perfect numbers, 248