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









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: The Riemann Hypothesis

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

: en

: /

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ed2k: ed2k stats

: 2007

: 576

: 04.06.2008

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$\xi$ function      14 1922 2527 31 38 50 203 313
Abelian character      490
Abelian integral      335
Accumulation point      see Condensation point
Adele ring      58 112
Adeles      106
Adjacency matrix      53
Adjacency matrix, graph      53
Adleman      512
Adleman Huang algorithm      130
Adleman Huang primality testing algorithm      512
Adleman Pomerance Rumely primality testing algorithm      512
Agrawal      63 509 511
Agrawal Kayal      114
Agrawal Kayal Saxena      114
Agrawal Kayal Saxena primality testing algorithm      63 114 509 511515
Agrawal Kayal Saxena primality testing algorithm, correctness proof      509 512 513 515 518 519 521
Agrawal Kayal Saxena primality testing algorithm, deterministic      63 509 511 513
Agrawal Kayal Saxena primality testing algorithm, polynomial time      63 509 511 512
Agrawal Kayal Saxena primality testing algorithm, running time      114 509 512 513
Agrawal Kayal Saxena primality testing algorithm, time complexity      509 512 513 519521
Agrawal Kayal Saxena primality testing algorithm, unconditional      63 509 511 512
Aithmetic variety      100
Aithmetic variety, L-functions      100
Aithmetical mean      351
Alacantara Bode      52
Algebraic closure      336
Algebraic curve      335
Algebraic curve, correspondence      335
Algebraic extension      336
Algebraic extension, degree      336
Algebraic extension, separable      336
Algebraic function      335
Algebraic function, finite field      335
Algebraic index theorem      105
Algebraic number field      58
Algebraic variety      102
Algebraic variety, cohomology      103
Algebraic variety, finite field      102 103
Algebraic variety, topological structure      103
Algebraically closed field      105
Algorithms      see Primality testing algorithms
Algorithms, alternating series      36
Algorithms, Euler MacLaurin summation      29 30
Algorithms, Euler MacLaurin summation, $\zeta(s)$      30
Algorithms, fast Fourier transform (FFT)      35
Algorithms, Fermats little theorem      511
Algorithms, Odlyzko Schonhage      35
Algorithms, Riemann hypothesis      29
Algorithms, Riemann Siegel      see Riemann Siegel formula
Algorithms, sieve of Eratosthenes      511
Algorithms, sieve of Eratosthenes, running time      511
Algorithms, Turing      34
Alternating zeta function      49 55 98 348 Zeta
Alternating zeta function, partial sum      348 (see also Partial sum)
Alternative hypothesis      133
American Institute of Mathematics      123
Analytic continuation      5 11 14 155 202 462 467 468
Analytic continuation, $L(s, \chi)$      112
Analytic continuation, $L(s, \chi_k)$      56
Analytic continuation, $L_p(s)$      57
Analytic continuation, $\dfrac{\zeta'(s)}{\zeta(s)}$      463
Analytic continuation, $\eta(s)$      56
Analytic continuation, $\Gamma(s)$      13
Analytic continuation, $\zeta(s)$      5 1113 37 55 58 97 99 111 123 202 322 461 464
Analytic continuation, $\zeta(s, a)$      144
Analytic continuation, L-function      100
Analytic continuation, Laplace transform      463
Analytical mean      381
Anzahl der Primzahlen      193 201
Approximate functional equation      32 33 126
Approximation, $N_0(T)$      152 165
Approximation, $N_M(T)$      165
Approximation, $S_M(T)$      156
Approximation, $\dfrac{\Gamma'(a)}{\Gamma(a)}$      251
Approximation, $\dfrac{\zeta'(s)}{\zeta(s)}$      252254
Approximation, $\log(\zeta(s))$      214
Approximation, $\mathfrak{R}(\dfrac{\Gamma'(a)}{\Gamma(a)})$      247
Approximation, $\mu(n)$      134
Approximation, $\pi(x)$      98100 124 140 171 173
Approximation, $\psi_{\mu}(x)$      230
Approximation, $\varphi(x)$      173 182184 187189
Approximation, $\zeta(s)$      32
Approximation, Diophantine      431
Approximation, distribution of prime numbers      509
Approximation, Li(x)      209
Approximation, N(T)      125 152 165
Approximation, S(T)      156
Approximation, Z(t)      162
Approximation, zero of $\zeta(s)$      44
Argument principle      19
Arithmetic functions, $(\dfrac{a}{n})$      512 (see also Jacobi symbol)
Arithmetic functions, $d_k(n)$      148 (see also General divisor function)
Arithmetic functions, $left(\dfrac{n}{p}\right)$      57 (see also Legendre symbol)
Arithmetic functions, $L_1(n)$      365 374
Arithmetic functions, $o_n(a)$      514
Arithmetic functions, $u_y(p)$      471
Arithmetic functions, $v_y(p)$      471
Arithmetic functions, $\chi_k(n)$      56
Arithmetic functions, $\Lambda(n)$      104 125 322 323 325 464 465 471
Arithmetic functions, $\mu(n)$      47 69 125 133 140 298 328 381 433 453 469 471 475 477
Arithmetic functions, $\omega(n)$      6 46
Arithmetic functions, $\phi(n)$      48 54 58 133 Eulers
Arithmetic functions, $\psi(n)$      514
Arithmetic functions, $\psi_{\nu}(n)$      220
Arithmetic functions, $\sigma(n)$      48 129
Arithmetic functions, $\sigma(n)$, upper bound      48
Arithmetic functions, $\sigma_k(n)$      48
Arithmetic functions, $\tau(n)$      130 381
Arithmetic functions, (a, n)      514
Arithmetic functions, a(k)      67
Arithmetic functions, A(n)      520 (see also Artins constant)
Arithmetic functions, h(d)      133 441
Arithmetic functions, L(n)      355 374
Arithmetic functions, log(n)      469
Arithmetic progression      134 381 392
Arithmetic progression, extended Riemann hypothesis      63
Arithmetic progression, prime numbers      63 111 133 233 241
Arithmetic progression, quadratic forms      233
Artin      64 103 114 130 520
Artin L-function      102 335 337
Artin L-function, polynomial      335
Artins constant      520 (see also Arithmetic functions A(n))
Artins primitive root conjecture      64 114 520
Artins primitive root conjecture, conditional proof      64
Artins primitive root conjecture, consequence      520
Artins primitive root conjecture, generalized Riemann hypothesis      64 520
Artins primitive root conjecture, Heath Brown      64
Artins primitive root conjecture, Murty Gupta      64
Artins primitive root conjecture, proof      64
Assertation de Riemann Stieltjes      224
Asymptotic behavior, $E_2(T)$      147
Asymptotic behavior, $F(\alpha)$      441
Asymptotic behavior, $H^{(n)}(z)$      26
Asymptotic behavior, $I_k(T)$      67
Asymptotic behavior, $M_{Z, f}(t)$      142 157
Asymptotic behavior, $N(\sigma, T + 1)  N(\sigma, T)$      65
Asymptotic behavior, $N_0(T)$      165
Asymptotic behavior, $S_1(T)$      66
Asymptotic behavior, $\Delta(t)$      158
Asymptotic behavior, $\Delta^{(k)}(t)$      158
Asymptotic behavior, $\dfrac1{\zeta(s)}$      325 327 331
Asymptotic behavior, $\dfrac{\zeta'(s)}{\zeta(s)}$      23 243 259
Asymptotic behavior, $\Gamma(s)$      21
Asymptotic behavior, $\Pi(x) - \pi(x)$      410
Asymptotic behavior, $\pi(x)$      18 98 241 298 320 321 392 463
Asymptotic behavior, $\psi(x)$      323 325 327 331 463 465
Asymptotic behavior, $\psi_{\mu}(x)$      230
Asymptotic behavior, $\Sigma_n\dfrac{\mu(n)}{n}$      302 480
Asymptotic behavior, $\Sigma_p\dfrac{\log(p)}{p-1}$      291 292
Asymptotic behavior, $\Sigma_p\dfrac{\log(p)}{p^s-1}$      259
Asymptotic behavior, $\Sigma_p\dfrac{\log(p)}{p}$      392 453 457 472 478 479
Asymptotic behavior, $\Sigma_p\log(p)$      289 379
Asymptotic behavior, $\Sigma_{p^m}\dfrac{\log(p)}{p^m}$      282 290 291
Asymptotic behavior, $\Sigma_{p^m}\log(p)$      282 286 289
Asymptotic behavior, $\Sigma_{\rho}\dfrac1{\gamma^2}$      407 408
Asymptotic behavior, $\Sigma_{\rho}\dfrac1{\gamma}$      407 408
Asymptotic behavior, $\theta(x)$      321
Asymptotic behavior, $\vartheta(t)$      25 26
Asymptotic behavior, $\vartheta(x)$      379 381 383 384 392395 397 398
Asymptotic behavior, $\vartheta^{(n)}(t)$      25 26
Asymptotic behavior, $\zeta(s)$      150
Asymptotic behavior, F(X, T)      485 488 490 497
Asymptotic behavior, H(z)      26
Asymptotic behavior, J(X, T)      502
Asymptotic behavior, Li(x)      98 240 241 297
Asymptotic behavior, M(x)      325 327 330 478 480483
Asymptotic behavior, mean square, S(t)      117
Asymptotic behavior, mean value, $\zeta(\frac12 + it)$      149
Asymptotic behavior, N(T)      15 41 140 498
Asymptotic behavior, prime numbers      225 231 233 235 239 240 274 317 319 461 489 520
Asymptotic behavior, S(T)      66
Asymptotic behavior, Sophie Germain primes      520
Asymptotic behavior, Z(t)      141 142
Asymptotic behavior, zeros, $R_n(s)$      372
Asymptotic behavior, zeros, $V_n(s)$      348 370
Asymptotic behavior, zeros, Davenport Heilbronn zeta function      144
Atkin      512
Atkinson      145
Atkins primality testing algorithm      512
Automorphic form      112 116 130
Automorphic function      5860
Automorphic representation      100
Automorphic representation, cuspidal      112
Auxiliary Tauberian theorem      462 468
Auxiliary Tauberian theorem, consequence      462 463
Auxiliary Tauberian theorem, prime number theorem      463
Auxiliary Tauberian theorem, proof      463 466468
Backlund      2931 34
Baez Duarte theorem      127
Balasubramanian      67 150
Balazard      127
Barnes      67 126 325
Barness G function      67
bays      44
Bell      169
Bernoulli number      30 158
Bernoulli polynomial      30
Bertrand      63 171
Bertrands postulate      63 171
Bertrands postulate, Riemann hypothesis      63
Bessel function      135
Beurling      52 53 127 381
Bieberbach conjecture      71
Bieberbach conjecture, proof      71
Bieberbach conjecture, proof, de Branges      71
binomial      57 465 517
Birch      134
Birch and Swinnerton-Dyer conjecture      134 135
Bohr      102 313 314 320 352 353
Bohrs theorem      314
Bombieri      95 97 109 111 139 144
Bombieri-A. Vinogradov theorem      115
Borwein      36
Bound      see Convexity bound Lower Upper
Bound, #Z^{(k)} (t)#      151
Bound, $E_1(T)$      147
Bound, $F(y, \lambda)$      424 425
Bound, $L(1, \chi_d)$      64
Bound, $L(\frac12 + it, \pi)$      113
Bound, $M_{Z,f}(t)$      164
Bound, $r_n(s)$      346 360 361
Bound, $R_{2v}$      30
Bound, $U_n(s)$      346
Bound, $V_n(s)$      371
Bound, $\dfrac{M(x)}{x}$      471 472
Bound, $\dfrac{\zeta'(s)}{\zeta(s)}$      230 254
Bound, $\pi(x)$      321
Bound, $\varphi(x)$      178
Bound, $\zeta(\frac12 + it)$      150
Bound, Dirichlet L-series      64
Bound, F(X, T)      490
Bound, order of $\zeta(s)$      146
Bound, R(T)      407
Bound, R(x)      385 386
Bound, series, nontrivial zeros      408
Bound, Skewes number      43 44
Bound, Z(t)      151
Bound, zero, imaginary part      254 257 258
Bound, zero, spacing      441
Brent      34
Breuil      101
Brownian bridge      127
Brownian motion      127
Brun      133
Brun Titchmarsh theorem      133
Burnol      127
Cahen      217 225 227 313 327
Cahens formula      313
Cardinality, $\mathfrak{G}$      517 518 521
Cardinality, $\mathfrak{G}$, lower bound      517 521
Cardinality, $\mathfrak{G}$, upper bound      518
Cardinality, G      517 518
Carleman      466
Carlemans formula      466
Carlson      102
Carlsons density theorem      102
Carlsons density theorem, Inghams exponent      102
Carmichael      511
Carmichael number      511
Cartier      129
Cauchy      38 163 313 323 447 462 466 481 484 499 505
Cauchy Schwarz inequality      163 447 481 484 505
Cauchys criterion      458
Cauchys estimate      371
Cauchys integral formula      38 101 462 466 499
Cauchys residue theorem      38
Cauchys Theorem      313 323 329
Central character      58 112
Central character, $\chi$      58
Central line      see Critical line
Cesaro      314
Cesaro mean      314 347 349
Chandrasekharan      462
CHARACTER      372 490
Character, $(\dfrac{a}{b})$      57
Character, $\chi$      58 112 372 373
Character, $\chi_3$      129 130
Character, $\chi_k$      56 132 133
Character, abelian      490
Character, central      112
Character, Dirichlet      64 100 111 130
Character, Dirichlet, $\chi$      111
Character, Dirichlet, $\chi_d$      64
Character, Dirichlet, nonprincipal      64
Character, Dirichlet, primitive      101
Character, primitive      56
Character, principal      56
Character, Tauberian      490
Characteristic polynomial      131
Characteristic polynomial, matrix      132
Characteristic polynomial, unitary matrix      67
Characteristic root      103
chebyshev      169 171 211 317 321 463 469
Chebyshev polynomial      36
Chebyshevs theorem      321
Chen      62
Chens theorem      62
Chowla      63 130
Class number      129 133 441
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