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, 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
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