Borwein P., Choi S., Rooney B. — The Riemann Hypothesis :: Электронная библиотека попечительского совета мехмата МГУ

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Borwein P., Choi S., Rooney B. — The Riemann Hypothesis

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Название: The Riemann Hypothesis

Авторы: Borwein P., Choi S., Rooney B.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 576

Добавлена в каталог: 04.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Series representation, $\mathfrak{R}(\log(\zeta(s)))$       16
Series representation, $\Pi(x)$       406 410
Series representation, $\psi(x)$       325 408 463 465 485
Series representation, $\psi_1(x)$       409
Series representation, $\theta(x)$       321
Series representation, $\vartheta(x)$       379 392
Series representation, $\xi(t)$       204
Series representation, $\zeta(s)$       33 464
Series representation, $\zeta(s, a)$       144
Series representation, F(X, T)      488
Series representation, L(x)      431
Series representation, Li(x)      98
Series representation, log(n)      469
Series representation, log(s - 1)      224
Series representation, M(x)      69 433 469 471 472 475 477
Series representation, R(t, x)      159
Series representation, S(t, x)      160
Series,       $\Sigma_{p^m}\log(p)$
Series, $\Sigma"$       266 278 306
Series, $\Sigma$       266 278 305
Series, $\Sigma'$       266 278 305
Series, $\Sigma\dfrac{y^{\sigma-1}}{\sigma^2+t^2}$       278
Series, $\Sigma_n\dfrac{\mu(n)}{n}$       298 480
Series, $\Sigma_n\dfrac{\mu(n)}{n}$ , convergence      298 453 457
Series, $\Sigma_p\dfrac{1}{p^s}$       214 224 320 322
Series, $\Sigma_p\dfrac{1}{p^s}$ , convergence      320
Series, $\Sigma_p\dfrac{\log(p)}{p-1}$       291
Series, $\Sigma_p\dfrac{\log(p)}{p}$       392 399 453 457 472 479
Series, $\Sigma_p\log(p)$       98 289 321 379 392
Series, $\Sigma_r$       275
Series, $\Sigma_{k=1}^n\dfrac{1}{k^s}$       339
Series, $\Sigma_{p^m}\dfrac{\log(p)}{p^m}$       282 290 291
Series, $\Sigma_{\rho}\dfrac{1}{(u-\rho)(1-\rho)}$       249—251
Series, $\Sigma_{\rho}\dfrac{1}{s-\rho}$       243 254—257 259 260 262 263 266—270 305 306
Series, $\Sigma_{\rho}\dfrac{1}{t^r}$       275
Series, $\Sigma_{\rho}\dfrac{1}{\gamma^2}$       407 408
Series, $\Sigma_{\rho}\dfrac{1}{\gamma}$       407 408
Series, $\Sigma_{\rho}\dfrac{1}{\rho(1-\rho)}$       245
Series, $\Sigma_{\rho}\dfrac{1}{\rho}$       244
Series, $\Sigma_{\rho}\dfrac{1}{\sigma^2+t^2}$       247 248
Series, $\Sigma_{\rho}\dfrac{y^{\rho-1}}{\rho(\rho-1)}$       290
Series, $\Sigma_{\rho}\dfrac{y^{\sigma-1}}{t^2}$       274 275 277
Series, $\Sigma_{\rho}\dfrac{y^{\sigma-1}}{\sigma^2+t^2}$       279—281 285 287 289 297
Series, $\Sigma_{\rho}\dfrac{\sigma}{\sigma^2+t^2}$       245
Series, Dirichlet      217 219 232 314
Series, Dirichlet,       221
Series, Dirichlet, sommablilite      314
Series, Farey      48
Series, Farey,       48
Series, Meyer      233
Series, nontrivial zeros      23 50 125 128 130 143 245 247—251 266—270 274 278 305 429 431 437 439 440 442 449 488
Series, upper bound      407
Series, Weber      233
Serre      111 114
Set, introspective integers      517
Set, N      514
Set, positive integers      514
Severi      105 335 336
Sherman      375
Shilov      151
Shimura      111
Shimura — Taniyama — Weil conjecture      111
Siegel      32 70 126 133 141 158
Sieve      64 133 475 477 478 512 “Upper
Sieve of Eratosthenes algorithm      511
Sieve of Eratosthenes algorithm, running time      511
Sieve, estimate      489
Sieve, large      475 477
Sieve, prime distribution      477
Sieve, prime number density      512
Sieve, technique      469
Sign,       433 435
Sign, $\lambda_n$       50
Sign, $\pi(x) - Li (x)$       43 142 209 403 405 410 415 418 426
Sign, $\pi(x) - Li (x)$ , change      405
Sign, $\xi(s)$       27
Sign, $\xi(t)$       101
Sign, $\xi(t)$ , change      101 102
Sign, $\xi(\hat{t})$       38
Sign, $\xi(\hat{t})$ , change      38
Sign, $\zeta(\frac12 + it)$       33
Sign, $\zeta(\frac12 + it)$ , change      33
Sign, change      357 358
Sign, Gauss sum      112
Sign, L(x)      431
Sign, L(x), verification      431
Simple pole,       221
Simple pole, $Z_2(\xi)$       147
Simple pole, $\Gamma(s)$       14
Simple pole, $\xi_k$       218
Simple pole, $\zeta(s)$       11 14 15 97 101 123 139 213 322 342
Simple pole, $\zeta(s, C)$       103
Simple pole, $\zeta(\sigma)$       17
Simple pole, Z(t, C)      103
Simple zero, $\zeta(s)$       14 38 39 102 141 142 440
Simple zero, Z(t)      141
Skewes      43 44 403 405
Skewes number      43
Skewes number, bound      43 44
Skewes number, upper bound      43 44
Snaith      67 115 131 132
Sobolev space      116
Solovay      63 512
Solovay — Strassen algorithm      see “Algorithms”
Solovay’s primality testing algorithm      512
Sophie Germain primes      512 520
Sophie Germain primes, asymptotic behavior      520
Sophie Germain primes, density      512 520
Sophie Germain primes, primality test      512 520
Soundararajan      67
Spacing between zeros, $\xi(t)$       105
Spacing between zeros, $\zeta(s)$       41 42 102 125 130 133 134 141 150 437 439 488 498
Spacing between zeros, $\zeta(s)$ , average      441
Spacing between zeros, $\zeta(s)$ , distribution      443
Spacing between zeros, $\zeta(s)$ , kth      41
Spacing between zeros, $\zeta(s)$ , normalized      41
Spacing between zeros, $\zeta(s)$ , small      43
Spacing between zeros, $\zeta_K(s)$       441
Spacing between zeros, $\zeta_K(s)$ , bound      441
Spacing between zeros, L-function      105 135
Spacing between zeros, Z(t)      142 143 150 157
Spacing between zeros, zeta function      105
Spectral interpretation, zeros, $\Lambda(s, \pi)$       115
Spectral interpretation, zeros, $\zeta(s)$       115 127 128 132
Spectral interpretation, zeros, L-function      116
Spectral theory      147 149
Spectral theory, non-Euclidean Laplacian      147
Spectrum      128
Spira      144
Steiner      193
Stern      193
Stieltjes      69 125 213 224 225 299
Stirling      21
Stirling’s formula      21 22 26 143 158
Strassen      63 512
Strassen’s primality testing algorithm      512
Subconvex estimate      115
Sudan      518
Sum of divisors function      47 48
Sum of divisors function, Riemann hypothesis      48
Summability      491
Summability, Dirichlet series      314
Support      54
Supremum, $M(n)n^{\frac12}$       70
Supremum, $M(n)n^{\frac12}$ , lower bound      70
Supremum, $\dfrac{M(x)}{x}$       471
Supremum, $\sigma_n$       241 258 261
Supremum, zeros, imaginary parts      255 258 441 443
surface      105
Surface, arithmetic hyperbolic      114

Surface, projective nonsingular      105
Swinnerton — Dyer      134
Sylvester      239
Symmetric functional equation      123 158
Symmetric functional equation, $\zeta(s)$       123 158
Symmetric group      53
Symmetric group, element      53
Symmetric group, element, order      53
Symmetric matrix      437 442
Symmetric matrix, eigenvalue      437 442
Symplectic matrix      105 132 437 442
Symplectic matrix, eigenvalue      105 437 442
Symplectic matrix, unitary      132
Szele Prize      339
Taniyama      111
Tate      106
Tauberian character      490
Tauberian theorem      327 (see also “Auxiliary Tauberian theorem”)
Tauberian theory      462
Taylor      101
Taylor series      16 26 323 351
Taylor’s Formula      152 159
Tchebotarev’s density theorem      100
Tchebychev      99 225 239 240 455 457
te Riele      44 62 69 70 102 125 141 142 147
Ternary additive divisor problem      149
Terras      129
Theoreme, Bohr      314
Theoreme, Cauchy      313
Theoreme, Halphen      225 231 232
Theoreme, nombres premiers      469 471
Theoreme, nombres premiers, demonstration      471
Theoreme, Riesz      314
Theoreme, von Mangoldt      274 275
Theorems, $I_1(T) \sim \logT$       67
Theorems, $I_1(T) \sim \logT$ , proof      67
Theorems, $I_2(T) \sim \dfrac1{2\pi^2}(log T)^4$       67
Theorems, $I_2(T) \sim \dfrac1{2\pi^2}(log T)^4$ , proof      67
Theorems, $I_k(T) \geq (a(k) + o(1))(\logT)^{k^2}$       67
Theorems, $I_k(T) \geq (a(k) + o(1))(\logT)^{k^2}$ , conditional proof      67
Theorems, $I_k(T) \geq (a(k) + o(1))(\logT)^{k^2}$ , proof      67
Theorems, $I_k(T) \geq 2(a_k + o(1))(\logT)^{k^2}$       67
Theorems, $I_k(T) \geq 2(a_k + o(1))(\logT)^{k^2}$ , proof      67
Theorems, $N(T) \sim \dfrac{T}{2\pi}log(\dfrac{T}{2\pi}) -\dfrac{T}{2\pi}$       15 19 24 31 41
Theorems, $N(T) \sim \dfrac{T}{2\pi}log(\dfrac{T}{2\pi}) -\dfrac{T}{2\pi}$ , proof      15 24 31
Theorems, $N(T) \sim \dfrac{T}{2\pi}log(\dfrac{T}{2\pi}) -\dfrac{T}{2\pi}$ , von Mangoldt      15 31
Theorems, $\dfrac1{\Gamma(\mu)}\Sigma_{p<x} \log(p)\log^{\mu-1}\left(\dfrac{x}{p}\right) \sim x      231
Theorems, $\pi(x) \sim Li (x)$       43 46 61 98 100 140 142 171 211 235 239—241 296 298
Theorems, $\pi(x) \sim Li (x)$ , proof      211 235
Theorems, $\pi(x) \sim \dfrac{x}{\log(x)}$       16 171 182 320 328 455 461 463
Theorems, $\psi(x) \sim x$       323 327 463
Theorems, $\psi(x) \sim x$ , proof      331 465
Theorems, $\Sigma_{n<x}\dfrac{\mu(n)}{n} =O(1)$       480
Theorems, $\Sigma_{n<x}\dfrac{\mu(n)}{n} =o(1)$ , proof      331
Theorems, $\Sigma_{n<x}\Lambda(n) \sim x$       125 325
Theorems, $\Sigma_{p<x}\dfrac{\log(p)}{p} = \log(x) + O(1)$       379
Theorems, $\Sigma_{p<x}\dfrac{\log(p)}{p} \sim log (x)$       292 392 453 472 479
Theorems, $\Sigma_{p<x}\dfrac{\log(p)}{p} \sim log (x)$ , proof      457 458
Theorems, $\Sigma_{p<x}\log(p) \sim x$       225 231 289
Theorems, $\Sigma_{p^m<x}\dfrac{\log(p)}{p^m} \sim log (x)$       291
Theorems, $\Sigma_{p^m<x}\log(p) \sim x$       289
Theorems, $\theta(x) \sim x$       321
Theorems, $\vartheta(x) \sim x$       379 381 392
Theorems, M(x) = o(x)      325 327 469 471 475 477
Theorems, M(x) = o(x), proof      330 469 471 475 477 478 480
Time magazine      70
Titchmarsh      58 63 65 66 102 127 133 311 325 417 462 498
Titchmarsh’s S(T) function      65
Trace formula, Selberg      128 129
Trace, Frobenius endomorphism      104
Transcendence of 2,      126
Trivial zero      14 97 124
Trivial zero, $\zeta(s)$       14 139 202 242 322 342
Tsang      150 156
Turan      71 142 339 341 433 478 483 484
Turan — Kubilius inequality      478 483 484
Turan — Kubilius inequality, dual      483 484
Turan — Kubilius inequality, large sieve inequality      478 483 484
Turan’s conjecture      433 524
Turan’s conjecture, conditional disproof      433
Turan’s inequality      433 (see also “Turan’s conjecture”)
Turan’s inequality, conditional disproof      433
Turan’s inequality, disproof      435
Turing      34 514
Turing machine      514
Turing machine, deterministic polynomial time      514
Turing’s algorithm      34
Twin prime constant      520
Twist      132
Twist, quadratic      132
Ungar      343
Ungar’s theorem      343
Unitary linear operator      437 442
Unitary matrix      67 105 129 437 442
Unitary matrix, characteristic polynomial      67
Unitary matrix, eigenvalue      105 129 437 442
Unitary matrix, symplectic      132
Unitary representation, irreducible      112
Upper bound sieve      478
Upper bound sieve, Selberg      478
Upper bound,       146
Upper bound,       146
Upper bound,       146 148
Upper bound,       66
Upper bound,       164
Upper bound, $\pi(x)$       98 99 321 465
Upper bound, $\pi_2(x,k)$       497
Upper bound, $\psi(x)$       99 408 465
Upper bound, $\sigma$       241 258 261 403 405 406 409
Upper bound, $\sigma(n)$       48
Upper bound, $\sigma(y)$       281 282
Upper bound, $\Sigma_{\rho}\dfrac{1}{\gamma^2}$       407
Upper bound, $\Sigma_{\rho}\dfrac{1}{\gamma}$       407
Upper bound, $\Sigma_{\rho}\dfrac{y^{\sigma-1}}{t^2}$       277 278
Upper bound, $\Sigma_{\rho}\dfrac{y^{\sigma-1}}{\sigma^2+t^2}$       279—281
Upper bound, $\varphi(x)$       178
Upper bound, $\vartheta(x)$       99 380 393
Upper bound, $\zeta(\frac12 + it)$       146
Upper bound,       148
Upper bound, cardinality of $\mathfrak{G}$       518
Upper bound, distribution of zeros      445
Upper bound, infimum, $M(n)n^{-\frac12}$       70
Upper bound, Li(x)      409 410
Upper bound, M(x)      478
Upper bound, prime distribution      477 478
Upper bound, S(T)      150
Upper bound, Skewes number      43 44
Upper bound, Z(t)      151
Upper bound, zeros, Davenport — Heilbronn zeta function      144
Upper bound, zeros, imaginary parts      449
Vaaler      496
Valeur principale, de Li(x)      240
Value,       38
Value, $\zeta'(s)$       49
Value, $\zeta(\frac12 + it)$       150
Value, S(T)      150
van de Lune      102 126 142
Van der Waerden      335
Variety, algebraic      102 103
Variety, arithmetic      100
Variety, field      105
Variety, finite field      105 115
Variety, finite field, Riemann hypothesis      105 115
Variety, general      131
Variety, general, Riemann hypothesis      131
Variety, non-singular      105
Variety, projective      105
Variety, zeta functions      105
Vaughan      487 504
Verdier      103

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