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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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Kossuth Prize      339
Kowalski      117
Kristensen      374
Kubilius      478 483 484
Kuznetsov      149
L-functions      97 100 101 111 121 129 325 Dirichlet Hasse
L-functions, $L(s, \chi)$      111 112 373
L-functions, $L(s, \chi)$, degree 1      112
L-functions, $L(s, \chi)$, partial sum      373
L-functions, $L(s, \chi)_1$, $\zeta(s)$      56
L-functions, $L(s, \chi_1)$      56
L-functions, $L(s, \chi_3)$      129
L-functions, $L(s, \chi_k)$      56 132
L-functions, $L(s,\Delta)$      112
L-functions, $L(s,\Delta)$, degree 2      112
L-functions, $L(s,\pi)$      59 113
L-functions, $L(s,\pi_p)$      59 112
L-functions, $L(s,\pi_{\infty})$      59 112
L-functions, $L_E(s)$      130
L-functions, $L_{\Delta}(s)$      130
L-functions, $\Lambda(s,\chi)$      112
L-functions, $\Lambda(s,\pi)$      59 113
L-functions, arithmetic variety      100
L-functions, automorphic      58-60
L-functions, automorphic form      130
L-functions, automorphic representation      100
L-functions, class      97
L-functions, degree m      112
L-functions, Dirichlet      111 132 373
L-functions, Dirichlet series      100
L-functions, elliptic curve      100 102 130 134 135
L-functions, elliptic curve, modularity      135
L-functions, elliptic curve, rational point      134
L-functions, evaluation, $\Delta$      286
L-functions, evaluation, $\Sigma_{p^m}\dfrac{\log(p)}{p^m}$      290
L-functions, evaluation, $\Sigma_{p^m}\log(p)$      282
L-functions, evaluation, $\Sigma_{\rho}\dfrac{1}{\sigma^2+t^2}$      247
L-functions, evaluation, $\Sigma_{\rho}\dfrac{y^{\sigma-1}}{t^2}$      274
L-functions, evaluation, $\Sigma_{\rho}\dfrac{y^{\sigma-1}}{\sigma^2+t^2}$      278
L-functions, evaluation, somme des p      274
L-functions, extremal      134
L-functions, family      105 131 135
L-functions, family, global      100
L-functions, family, Maass waveform      101 106
L-functions, family, modular      102 130 134
L-functions, family, Ramanujan r cusp form      132
L-functions, k-tuple correlation      442
L-functions, k-tuple correlation function      442
L-functions, k-tuple correlation, zeros of $\zeta(s)$      442
L-functions, k-tuple hypothesis      490
L-functions, kth consecutive spacing      41
L-functions, L-series      see Dirichlet L-series
L-functions, nth divided difference function      154
L-functions, nth harmonic number      48
Lacroix      252
Lagarias      48 50 129
Landau      6 7 97 102 133 147 311 313 319 352 356 364 453 455 457 462
Landau Siegel zero      133 134
Landaus classical result      147
Landaus paradigma      364
Landaus theorem      356 357 364
Langlands      59 113 116
Langlands conjecture      524
Langlands program      130
Laplace      128 459 462 463
Laplace integral      459 462
Laplace operator      128
Laplace operator, eigenvalue      128
Laplace transform      462 463
Large sieve      475 477
Large sieve inequality      475 478 483 484
Large sieve inequality, Elliott      478 483
Large sieve inequality, prime number theorem      475
Large sieve inequality, proof      478 483
Large sieve inequality, Turan Kubilius inequality      478 483 484
Large sieve, exponential sum      483
Large sieve, Montgomery      478
Large sieve, prime number theorem      475 477
Large sieve, standard version      483
Laurent series      17
Lavrik      151
LCM      see Least common denominator
LCM(m)      see Least common denominator
Least common denominator      514
Lebesgue      412 467
Lebesgue integral      412
Leech      432
Lefschetz fixed point formula      104
legendre      124 173 183 185 189 192 252 321 322 461
Legendre symbol      57 (see also Arithmetic functions
Legendres conjecture      see Conjectures
Legendres formula      183 185 189
Lehman      34 44 142
Lehmer      43 432
Lehmers phenomenon      43 139 142 144
Lejeune      239
Lenstra      517 521
Levinson      27 102 126
lh      see Lindelof hypothesis
Li      71
Limite inferieure, $1 - \sigma_n$      241 261 265 270
Limite superieure, $\sigma_n$      241 258 261
Lindelof      65
Lindelof hypothesis      65 126 145 146
Lindelof hypothesis, conditional disproof      149
Lindelof hypothesis, equivalent statement      65 146 148 149
Lindelof hypothesis, grand      see Grand Lindelof hypothesis
Lindelof hypothesis, implication      146
Lindelof hypothesis, Riemann hypothesis      65 126 145 146
Lindelof hypothesis, statement affecting      148
Lindelof hypothesis, statement implying      65
Lindelof hypothesis, zeros of $\zeta(s)$      146
Line of symmetry      see Critical line
Linear operator      105 437 442
Linear operator, eigenvalue      105 437 442
Linear operator, Hermitian      437 442
Linear operator, self-adjoint      105
Linear operator, unitary      437 442
Linear operator, zeros of $\zeta(s)$      437 442
Linear transformation, eigenvalue      115
Liouvilles function      6 8 46 353 429 431
Liouvilles function, prime number theorem      7
Liouvilles function, Riemann hypothesis      6 46
Lipschitz      222
Littlewood      43 51 62 64 67 100 114 126 129 131 142 146 311 325 327 403 405407 419
Littlewoods theorem      405
Local representation      58 112
Location of prime numbers      63
Location of zeros, $(1 - 2^{1-2s})\zeta(2s)$      314
Location of zeros, $J_n(s)$      352
Location of zeros, $L(s, \chi)$      373
Location of zeros, $L_{\nu}(s)$      222
Location of zeros, $M_{Z, f}(t)$      152
Location of zeros, $U_n(s)$      345 358
Location of zeros, $V_n(s)$      366 372
Location of zeros, $\Lambda(s)$      111
Location of zeros, $\Lambda(s, \chi)$      116
Location of zeros, $\Lambda(s, \pi)$      115
Location of zeros, $\xi(s)$      38
Location of zeros, $\xi(t)$      204
Location of zeros, $\zeta(s)$      14 15 24 30 70 97 101 102 116 125 126 139 140 202 211 213 242 254 258 274 311 313 322 342 343 403 405407 409 429 432 434 435 445
Location of zeros, Davenport Heilbronn zeta function      144
Location of zeros, L-functions      100
Location of zeros, partial sum      71 339
Location of zeros, Z(t)      152
Locus      335
Lois asymptotiques relatives aux nombre premiers      274
Lower bound, $1 - \sigma$      241 261 265 270 308 309
Lower bound, $E_1(T)$      146
Lower bound, $E_2 (T)$      146
Lower bound, $L(1, \chi_k)$      133
Lower bound, $o_n(a)$      515
Lower bound, $\gamma_n$      243
Lower bound, $\Lambda$      51
Lower bound, $\pi(x)$      98 99 321
Lower bound, $\psi(x)$      99
Lower bound, $\Sigma_p\dfrac{\log(p)}{p}$      399 401
Lower bound, $\Sigma_{\rho}\dfrac{1}{s-\rho}$      257
Lower bound, $\varphi(x)$      178
Lower bound, $\vartheta(x)$      99 380
Lower bound, cardinality of $\mathfrak{G}$      517 521
Lower bound, class number      134
Lower bound, Davenport Heilbronn zeta function      144
Lower bound, Dirichlet polynomial      127
Lower bound, F(X, T)      497
Lower bound, h(d)      133 134
Lower bound, imaginary parts      308 449
Lower bound, L(n)      355
Lower bound, LCM(m)      514
Lower bound, product of prime numbers      521
Lower bound, S(t)      116
Lower bound, supremum, $M(n)n^{\frac12}$      70
Lower bound, Z(t)      151
Lower bound, zeros, $\zeta(s)$      144 450
Lucas      319
Maass L-function      102
Maass waveform      101 106
Maass Selberg formula      116
Maclaurin      29
MacLaurin summation formula      101
Mansion      238 239 309
Matrix, adjacency      53
Matrix, eigenvalue      40 105 115
Matrix, hermitian      130 131 437 442 443
Matrix, orthogonal      105 132
Matrix, random      40 42 109
Matrix, Redheffer      52
Matrix, Redheffer, determinant      52
Matrix, symmetric      437 442
Matrix, symplectic      105 132 437 442
Matrix, unitary      67 105 129 437 442
Matrix, unitary symplectic      132
Maximum, $\xi(t)$      40 102
Maximum, $\zeta(\frac12 + it)$      150
Maximum, Z(t)      142144
Mean      351
Mean square, $\zeta(\frac12+ it)$      164
Mean square, S(t)      117
Mean square, S(t), asymptotic behavior      117
Mean value formula      142 145
Mean value formula, critical line      145
Mean value theorem      17 154
Mean value theorem, Wirsing      477
Mean value, $M_{Z,f}(t)$      152 163
Mean value, $\zeta(s)$      66
Mean value, $\zeta(\frac12+ it)$      149
Mean value, $\zeta(\frac12+ it)$, asymptotic behavior      149
Mean value, Z(t)      152
Mean, $C_n(s)$      347 349
Mean, $E_1(T)$      148
Mean, $E_2(T)$      148
Mean, $J_n(s)$      352
Mean, $R_n(s)$      347 349
Mean, analytical      381
Mean, arithmetical      351
Mean, Cesaro      314 347 349
Mean, Jensen      352 375
Mean, Riesz      347 349
Measure      52 54
Measure, discrete      54
Measure, ergodic      52
Mehta      42
mellin      313
Mellin inversion formula      25
Mellin transform      104 462 463
Meromorphic function      19 59 147
Mertens      37 69 141 298 299 433 464 477 479
Mertens conjecture      69 125 141 433 435
Mertens conjecture, conditional disproof      433
Mertens conjecture, disproof      69 125
Mertens conjecture, disproof, Odlyzko te Riele      69
Mertens conjecture, Riemann hypothesis      69
Mertens formula      479
Mertens function      47 48 69 433 475 M(x))
Mertens function, order      69 70
Mertens function, Riemann hypothesis      47
Mertens theory      477
Meyer      233
Meyer series      233
Millennium Prize      4 72 95 111 117
Miller      63 114 432 512
Millers primality testing algorithm      100 114 512
Millers primality testing algorithm, running time      114
Minimum, $\xi(t)$      40 102
Minimum, $\zeta(\frac12+ it)$      150
Minimum, Z(t)      142144
Mobius function      47 69 125 133 134 140 298 381 433 453 469 471 475 477 \mu(n))
Modular form      111 130
Modular function      100
Modular group      112 147
Module      336
Module, correspondence      336
Modulus      64
moment      134 135
Moment, $L(\frac12+it,\pi)$      115
Moment, $L(\frac12+it,\pi)$, asymptotics      115
Moment, $L(\frac12,\pi)$      115
Moment, $L(\frac12,\pi)$, asymptotics      115
Moment, $\zeta(s)$      66 131
Moment, $\zeta(\frac12 + it)$      126 139 145 148 149
Moment, Brownian bridge      127
Moment, characteristic polynomial      131 132
Moment, family of L-functions      132
Moment, zeta function      137
Monodromy group      115
Montgomery      4042 71 102 105 130 131 134 150 339 437 439 478 485 487 488 498 502
Montgomery Odlyzko law      42 115
Montgomerys conjecture      see Conjectures
Montgomerys large sieve      478
Montgomerys pair correlation, conjecture      41 43
Montgomerys pair correlation, conjecture, Riemann hypothesis      41
Motohashi      133 146149
Motohashis fundamental explicit formula      147
Moyenne de Cesaro      314
Mueller      488 489 498
Multiple pole, $Z_2(\xi)$      147
Multiple zero, $\xi(t)$, odd      38
Multiple zero, L-function      134 135
Multiple zero, L-function, elliptic curve      135
Multiple zero, L-function, triple      134
Multiplicity      see Order
Multiplicity, $p_k$      155
Multiplicity, pole, $L_1(s)$      221
Multiplicity, pole, $Z_2(\xi)$      147
Multiplicity, pole, $\xi_k(s)$      218
Multiplicity, pole, $\zeta(s)$      11 15 97 101 123 139 213 322 342
Multiplicity, pole, $\zeta(s, C)$      103
Multiplicity, pole, Z(t, C)      103
Multiplicity, zero, $M_{Z,f}(t)$      152
Multiplicity, zero, $\xi(t)$      101
Multiplicity, zero, $\xi(\hat{t})$      38
Multiplicity, zero, $\zeta(s)$      39 102 128 134 141 142 431 440442 449
Multiplicity, zero, elliptic curve L-function      135
Multiplicity, zero, L-function      134
Multiplicity, zero, Z(t)      141 151 152
Newman      51 61 127 453 455 459 461463
Non-Euclidean Laplacian      147
Non-Euclidean Laplacian, discrete spectrum      147
Nonsingular model      336
Nontrivial zero      5 15 102 241
1 2 3 4 5 6 7 8 9
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