Index

Ψ(x, y), 11, 115, 206

upper bound for Ψ(x, log x), 257

γ, 98

Li(x), 86

and , xiii

ω(n) and Ω(n), 111

π(x), 1

σ−1(n), 254

∼, xiii

ζ(s), 4

eπ

√

163,

32

f-nomial period, see also Gaussian period

m-gonal number, 148

s(n), 248

abundant numbers, 248

density, 251

aliquot sequence, 252

geometric growth, 263, 268

almost prime, 168

amicable numbers, 252

Artin’s constant, 245

Bertrand’s postulate, 89, 94, 108

Besicovitch set, 276

big-Oh notation, xii, xiii

Bombieri–Vinogradov theorem, 109

Bonferroni inequalities, 177

Brun’s constant, 180

Brun’s method, 206

Brun’s pure sieve, 168, 175

application to estimating π2(x), 179

general version, 178

working version, 178

Brun–Hooley sieve, 168, 182

application to sums of primes

(Schnirelmann’s theorem), 196

application to the generalized twin prime

problem, 190, 196

application to the Goldbach problem,

185, 193

lower bound method, 191

upper bound method, 183

Brun–Titchmarsh inequality, 206, 245, 273

Carath´ eodory’s theorem, 156

Cardano’s formula, 71

Catalan–Dickson conjecture, 253

character of a finite abelian group, 123

characters of

(Z/mZ)×,

124

classification of characters, 124

orthogonality relations, 125, 142

trivial character, 123

Chebotarev density theorem, 26, 40

Chebyshev’s theorems, 89, 92, 217, 220

class number 1 problem, 22

cluster prime, 204

composite numbers, 1

of the form α · n! + 1, 31

of the form

ξαn

, 33

constructibility

of regular 17-gon, 45, 56, 78

of regular n-gon (Gauss–Wantzel

characterization), 46, 55

rudiments, 50

constructible number, 51, 77

cubic reciprocity law (Eisenstein), 75

cubic reciprocity law (Jacobi), 50, 64, 70,

82

form of Z.-H. Sun, 73

cubic residuacity

301