Index 303

Euler’s form of odd perfect numbers, 250

heuristic argument suggesting only

finitely many odd examples, 258

Wirsing’s theorem, 251, 267

proof of, 255

polygonal number theorem, 148

prime number graph, 111

prime number theorem

as a consequence of the Wiener–Ikehara

theorem, 214

discovery by Gauss, 86

equivalence to the nonvanishing of ζ(s)

on (s) = 1, 215, 238

equivalent forms in terms of θ and ψ, 90

error term, 105

for arithmetic progressions, 101, 240, 245

for polynomials, 115

prime numbers

definition, 1

divergence of reciprocal sum, 7, 10, 173

Fermat prime, 30

have density zero, 88

heuristics from probability, 100

infinitude

Braun, M´ etrod, 2

Erd˝ os, 10

Euclid, 2

Euler (1st proof), 7

Euler (2nd proof), 2

Furstenberg, 12

Goldbach, 3

Hacks, 8, 36

Hemminger, 4

Perott, 9

Saidak, 4

Stieltjes, 2

Washington, 13

Wunderlich, 4

Mersenne prime, 29, 103, 249

of the form

1

2

(n2

+ 1), 42

of the form αn , 33

of the form n ·

2n

+ 1, 203

of the form

n2

+ 1, 28, 172

of the form

n2

+ k, 110

polynomial with prime positive range, 32

prime producing machine, 13

principle of inclusion-exclusion, 170

pseudoperfect number, 276

quadratic reciprocity, 46

cyclotomic proof, 61, 63

first supplementary law, 63

second supplementary law, 64

Rabinowitsch’s theorem, 15

random sieve (Hawkins), 104

Riemann Hypothesis, 105

connection with large values of σ(n), 269

Riemann zeta function

continuation to (s) 0, 214

definition, 5

Euler factorization, 5

evaluation of ζ(2) and ζ(4), 35

nonvanishing on (s) = 1, 238

Romanov’s theorem, 210

Schnirelmann density, 197

Schnirelmann’s theorem, 196

Selberg’s fundamental formula, 215

for arithmetic progressions, 241

proof of, 221

set of multiples, 276

sieve of Eratosthenes, 163

sieve of Eratosthenes–Legendre, 169

application to estimating π(x), 165

general version, 170

Legendre’s formula, 164

sign changes of π(x) − Li(x), 106

smooth numbers, 11, 115

sociable numbers, 253, 263

distribution of, 253, 266

squarefull number, 36, 145

sums of three primes, 201

sums of three squares, 134

number of representations, 140

sums of two squares (number of

representable integers), 174

superperfect number, 270

them there hills, xi

twin primes, 27

convergence of reciprocal sum, 168, 179

infinitude of almost prime pairs, 168, 196

twin prime conjecture (qualitative), 27

twin prime conjecture (quantitative), 102

twin prime conjecture for polynomials,

116

untouchable number, 272

Vinogradov’s three primes theorem, 201

Waring’s problem, 151

determination of g(k), 161

finiteness of g(k), 152

upper bound on G(k), 162

weird number, 277

Wiener–Ikehara theorem, 214

zeta function, see also Riemann zeta

function