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
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
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
Previous Page Next Page