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References
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Weil, A., Sur les fonctions algebriques a corps de constantes fini, C. R. Acad. Sci.Paris 210 (1940), 592–594.
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500 References
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Widder, D. V., The Laplace Transform, Princeton Mathematical Series, v. 6, Prince-ton University Press, Princeton, N. J., 1941.
Wiener, N., Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100.Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141
(1995), no. 3, 443–551.
Index
L-functions, 95, 98, 99, 107, 116, 123,307, see Artin L-function, seeDirichlet series, see Hasse–WeilL-function
L(s,∆), 108
degree 2, 108
L(s, χ), 107, 108, 350
degree 1, 108
partial sum, 350
L(s, χ1), 56
ζ(s), 56
L(s, χ3), 123
L(s, χk), 56, 126
L(s, π), 59, 109
L(s, π∞), 59, 108
L(s, πp), 59, 108
LE(s), 124
L∆(s), 124
Λ(s, χ), 108
Λ(s, π), 59, 109
arithmetic variety, 98
automorphic, 58–60
automorphic form, 124
automorphic representation, 98
class, 95
degree m, 108
Dirichlet, 107, 126, 350
Dirichlet series, 98
elliptic curve, 98, 100, 124, 128, 129
modularity, 129
rational point, 128
extremal, 128
family, 103, 125, 129
global, 98Maass waveform, 99, 104modular, 100, 124, 128Ramanujan τ cusp form, 126
L-series, see Dirichlet L-seriesk-tuple correlation, 409
zeros of ζ(s), 409k-tuple correlation function, 409k-tuple hypothesis, 451kth consecutive spacing, 41nth divided difference function, 146nth harmonic number, 48evaluation∆, 272∑
ρ1
σ2+t2, 233∑
ρyσ−1
σ2+t2, 264∑
ρyσ−1
t2, 260∑
pmlog(p)
pm , 276∑pm log(p), 268
somme des ρ, 260
Abelian character, 451Abelian integral, 314accumulation point, see condensation
pointadele ring, 58, 108adeles, 104adjacency matrix, 53
graph, 53Adleman, 471Adleman–Huang algorithm, 124Adleman–Huang primality testing
algorithm, 471
502 Index
Adleman–Pomerance–Rumely primalitytesting algorithm, 471
Agrawal, 63, 469, 470Agrawal–Kayal, 110Agrawal–Kayal–Saxena, 110Agrawal–Kayal–Saxena primality
testing algorithm, 63, 110,469–474
correctness proof, 469, 471, 472, 474,477, 478, 480
deterministic, 63, 469, 470, 472polynomial time, 63, 469–471running time, 110, 469, 471, 472time complexity, 469, 471, 472,
478–480unconditional, 63, 469–471
Alacantara–Bode, 52algebraic closure, 315algebraic curve, 314
correspondence, 314algebraic extension, 315
degree, 315separable, 315
algebraic function, 314finite field, 314
algebraic index theorem, 103algebraic number field, 58algebraic variety, 100
cohomology, 101finite field, 100, 101topological structure, 101
algebraically closed field, 103algorithms, see primality testing
algorithmsalternating series, 36Euler–MacLaurin summation, 29, 30ζ(s), 30
fast Fourier transform (FFT), 35Fermat’s little theorem, 470Odlyzko–Schonhage, 35Riemann hypothesis, 29Riemann–Siegel, see Riemann–Siegel
formulasieve of Eratosthenes, 470
running time, 470Turing, 34
alternating zeta function, 49, 55, 96,325, see Dirichlet eta function, seezeta functions, (1− 21−s)ζ(s)
partial sum, 325, see partial sumalternative hypothesis, 127American Institute of Mathematics, 117analytic continuation, 5, 11, 14, 147,
191, 426, 431, 432L-function, 98L(s, χ), 108L(s, χk), 56Lp(s), 57Γ (s), 13η(s), 56ζ′(s)ζ(s)
, 427
ζ(s), 5, 11–13, 37, 55, 58, 95, 97, 107,117, 191, 304, 425, 428
ζ(s, a), 136Laplace transform, 427
analytical mean, 356Anzahl der Primzahlen, 183, 190approximate functional equation, 32,
33, 120approximationN(T ), 119, 144, 157N0(T ), 144, 157NM (T ), 157S(T ), 148SM (T ), 148Z(t), 154
<( Γ ′(a)Γ (a)
), 233Γ ′(a)Γ (a)
, 237ζ′(s)ζ(s)
, 238–240
log(ζ(s)), 201µ(n), 128π(x), 96–98, 118, 132, 162, 163ψµ(x), 217ϕ(x), 163, 172–174, 177–179ζ(s), 32Li(x), 198Diophantine, 400distribution of prime numbers, 469zero of ζ(s), 44
argument principle, 19arithmetic functions
( an
), 471, see Jacobi symbol(a, n), 473A(n), 479, see Artin’s constantL(n), 332, 351L1(n), 342, 351
Index 503
Λ(n), 102, 119, 304, 305, 307, 428,429, 434, see von Mangoldt’sfunction
χk(n), 56
λ(n), 6, 8, 46, 330, 399, 400, seeLiouville’s function(
np
), 57, see Legendre symbol
log(n), 433
µ(n), 47, 69, 119, 127, 132, 284, 309,356, 402, 419, 433, 434, 438, 439,see Mobius function
ω(n), 6, 46
φ(n), 48, 54, 58, 127, see Euler’sphi function, see Euler’s totientfunction
ψ(n), 473
ψν(n), 207
σ(n), 48, 123
upper bound, 48
σk(n), 48
τ(n), 124, 356, see Ramanujan’s taufunction
a(k), 67
dk(n), 140, see general divisorfunction
h(d), 127, 408, see class number
on(a), 473
uy(p), 434
vy(p), 434
arithmetic progression, 128, 356, 365
extended Riemann hypothesis, 63
prime numbers, 63, 107, 127, 220, 227
quadratic forms, 220
arithmetic variety, 98
L-functions, 98
arithmetical mean, 328
Artin, 64, 101, 110, 124, 479
Artin L-function, 100, 314, 316
polynomial, 314
Artin’s constant, 479, see arithmeticfunctions, A(n)
Artin’s primitive root conjecture, 64,110, 479
conditional proof, 64
consequence, 479
generalized Riemann hypothesis, 64,479
proof, 64
Heath-Brown, 64Murty–Gupta, 64
assertation de Riemann–Stieltjes, 211asymptotic behaviorE2(T ), 139F (X,T ), 447, 449, 451, 458F (α), 408H(z), 26H(n)(z), 26Ik(T ), 67J(X,T ), 463M(x), 307, 308, 311, 440, 442–445MZ,f (t), 134, 149N(T ), 15, 41, 132, 459N(σ, T + 1)−N(σ, T ), 65N0(T ), 157Z(t), 133, 134∆(t), 150∆(k)(t), 150Γ (s), 21Π(x)− π(x), 381
1ζ(s)
, 307, 308, 312ζ′(s)ζ(s)
, 23, 229, 245
S(T ), 66S1(T ), 66π(x), 18, 96, 227, 284, 302, 303, 365,
427ψ(x), 305, 307, 308, 312, 427, 429ψµ(x), 217∑
ρ1
γ2 , 378, 379∑ρ
1γ
, 378, 379∑n
µ(n)n
, 288, 442∑p
log(p)p−1
, 277, 278∑p
log(p)ps−1
, 245∑p
log(p)p
, 365, 419, 422, 435, 440, 441∑p log(p), 275, 354∑pm
log(p)pm , 268, 276, 277∑
pm log(p), 268, 272, 275θ(x), 303ϑ(t), 25, 26ϑ(x), 354, 356, 358, 359, 365–368,
370, 371ϑ(n)(t), 25, 26ζ(s), 142Li(x), 96, 226, 227, 283mean squareS(t), 113
504 Index
mean valueζ( 1
2+ it), 141
prime numbers, 212, 218, 220, 222,225, 226, 260, 300, 301, 425, 450,479
Sophie Germain primes, 479zerosRn(s), 349Vn(s), 325, 347Davenport–Heilbronn zeta function,
136Atkin, 471Atkin’s primality testing algorithm, 471Atkinson, 137automorphic form, 108, 112, 124automorphic function, 58–60automorphic representation, 98
cuspidal, 108auxiliary Tauberian theorem, 426, 432
consequence, 426, 427prime number theorem, 427proof, 427, 430–432
Backlund, 29–31, 34Baez–Duarte theorem, 121Balasubramanian, 67, 142Balazard, 121Barnes, 67, 120, 307Barnes’s G function, 67Bays, 44Bell, 161Bernoulli number, 30, 150Bernoulli polynomial, 30Bertrand, 63, 162Bertrand’s postulate, 63, 162
Riemann hypothesis, 63Bessel function, 129Beurling, 52, 53, 121, 356Bieberbach conjecture, 71
proof, 71de Branges, 71
binomial, 57, 429, 476Birch, 128Birch and Swinnerton-Dyer conjecture,
128, 129Bohr, 100, 297, 298, 302, 329, 330Bohr’s theorem, 298Bombieri, 94, 95, 106, 107, 131, 136Bombieri–A. Vinogradov theorem, 111
Borwein, 36bound, see convexity bound, see lower
bound, see upper boundE1(T ), 139F (X,T ), 451F (y, λ), 395, 396L(1, χD), 64
generalized Riemann hypothesis, 64L( 1
2+ it, π), 109
MZ,f (t), 156R(T ), 378R(x), 360, 361R2v, 30Un(s), 323Vn(s), 348Z(t), 143Z(k)(t), 143M(x)
x, 434, 435
ζ′(s)ζ(s)
, 217, 240
π(x), 303ϕ(x), 168ζ( 1
2+ it), 142
rn(s), 323, 337, 338Dirichlet L-series, 64order of ζ(s), 138series
nontrivial zeros, 379Skewes number, 43, 44zero
imaginary part, 240, 243, 244spacing, 408
Brent, 34Breuil, 99Brownian bridge, 121Brownian motion, 121Brun, 127Brun–Titchmarsh theorem, 127Burnol, 121
Cahen, 204, 212, 214, 297, 308Cahen’s formula, 297cardinalityG, 476, 477G, 476, 477, 480
lower bound, 476, 480upper bound, 477
Carleman, 430Carleman’s formula, 430
Index 505
Carlson, 100Carlson’s density theorem, 100
Ingham’s exponent, 100Carmichael, 470Carmichael number, 470Cartier, 123Cauchy, 38, 155, 297, 305, 414, 426, 430,
443, 446, 460, 466Cauchy’s criterion, 423Cauchy’s estimate, 348Cauchy’s integral formula, 38, 99, 426,
430, 460Cauchy’s residue theorem, 38Cauchy’s theorem, 297, 305, 310Cauchy–Schwarz inequality, 155, 414,
443, 446, 466central character, 58, 108χ, 58
central line, see critical lineCesaro, 298Cesaro mean, 298, 324, 326, see meanChandrasekharan, 426character, 349, 451
( ab), 57
χ, 58, 108, 349, 350χ3, 123, 124χk, 56, 126, 127Abelian, 451central, 108Dirichlet, 64, 98, 107, 124χ, 107χD, 64nonprincipal, 64primitive, 99
primitive, 56principal, 56Tauberian, 451
characteristic polynomial, 125matrix, 126unitary matrix, 67
characteristic root, 101Chebyshev, 161, 162, 199, 300, 303, 427,
433Chebyshev polynomial, 36Chebyshev’s theorem, 303Chen, 62Chen’s theorem, 62Chowla, 63, 124class number, 123, 127, 408
L-functions, 127lower bound, 128
class number formula, 127class number problem, 124, 127class of functions, 71Eβ
α, 143L-functions, see L-functionsSβ
α, 143T β
α , 154convolution function, 143
Clay Mathematics Institute, 4, 93, 94,106, 107
prize, 131closed orbit, 52clustering point, see condensation pointco-NP, see complexity classcohomology, 101, 103
algebraic variety, 101zeta functions, 103
cohomology group, 111Frobenius endomorphism, 111
coincidence number, 314compact support, 54complex integration theory, 426complex zero, see nontrivial zerocomplexity class, 470
co-NP, 471, 473NP, 471, 473P, 469, 470, 473
complexity theory, 470composite integer, 470computationA∗T (u), 399, 402, 404L(n), 351N(T ), 34, 35N1(T ), 38S(T ), 142Z(t), 32, 33γn, 401
table, 403, 404π(x), 46
Gauss, 46ρn, 99, 133, 399ζ′(ρn), 401ζ(s), 29, 33, 36, 399, 401
table, 401zeroζ′(s), 38ζ(s), 31, 32, 35, 38, 39, 70, 125
506 Index
Artin L-function, 100Dirichlet L-function, 100elliptic curve L-function, 100Maass L-function, 100
Comrie, 388condensation point, 322, 326–328, 352conductor, 109, 110
curve, 124elliptic curve, 110representation, 109
congruence test, 110conjecturesF (X,T ) ∼ T
2πlog(T ), 447, 449
Ik(T ) ∼ a(k)g(k)
Γ (k2+1)(log T )k2
, 67
Ik(T ) ∼ ck(log T )k2, 67
π(x) < Li(x), 43, 134, 198disproof, 43, 44, 376false, 134Skewes, 44
π(x) ∼∫ x
2dt
log(t), 118, 119
π(x) ∼ Li(x), see theoremsArtin’s primitive root, 64, 110, 479
conditional proof, 64consequence, 479generalized Riemann hypothesis,
64, 479proof, 64
Bieberbach, 71de Branges, 71proof, 71
Birch and Swinnerton-Dyer, 128, 129density of prime numbers, 471Gauss, 118, 119
proof, 127restatement, 119
Goldbach, 62, 301Goldbach (strong), 62Goldbach (weak), 62
generalized Riemann hypothesis, 62statement implying, 62
GUE, 125Hilbert–Polya, 37, 40Keating–Snaith, 67, 111, 125, 126Langlands, 59Mertens, 37, 69, 119, 133, 402, 404
conditional disproof, 402disproof, 69, 119
moments of L-functions, 126
Montgomery’s pair correlation, 41,43, 124, 128, 405, 449
equivalent statement, 463Montgomery–Odlyzko law, 42Polya, 399, 400
disproof, 399–402Riemann hypothesis, 400, 401verification, 399
pair correlation, 449Shimura–Taniyama–Weil, 107Sophie Germain prime density, 471,
479Turan, 402
conditional disproof, 402Weil, 106, 111, 313
conjugacy class, 137norm, 137
Connes, 112Conrad, 99Conrey, 27, 67, 71, 100, 113, 116, 117,
125, 127, 131constant, see Euler’s constantBn, 30, 150G, 377, 380Hn, 48R2v, 30, 31Λ, 51
lower bound, 51γ, see Euler’s constantλ(n)
Riemann hypothesis, 50sign, 50
λn, 50, 122ρ, 163dk, 36g, 101, 315gn, 33, 34de Bruijn–Newman, 51twin prime, 479
constante d’Euler, 230, 434contour integral, 38, 146, 147, 191, 193,
205, 206, 213, 215, 217, 219, 305,392, 419, 420, 430, 431
classical, 420finite, 419, 420infinite, 420
contragredient representation, 59, 109convergence theorem
Ingham, 420
Index 507
convexity bound, 120convolution, 434
Dirichlet, 434convolution functions, 131, 143, 149,
154MZ,f (t), 134, 143, 146
M(k)Z,f (t), 143
class, 143correspondence, 314
algebraic curve, 314complementary, 315curve, 101degree, 314, 315irreducible, 314module, 315product, 315
critical line, 15, 30, 34, 49, 107, 119,132, 133, 136–138, 142, 157, 209,296, 320, 407, 426
L-function, 60, 126, 129Λ(s, χ), 112η(s), 49ζ(s), 5, 15, 18, 24, 27, 30–32, 34, 35,
38, 40, 100, 112, 119, 120, 125,132–134, 157, 212, 240, 244, 296,297, 307, 400, 407
ζK(s), 408Davenport–Heilbronn zeta function,
136large value, 134mean value formula, 137zeros, 100, 111, 112, 119, 120, 125,
126, 129, 132–134, 136, 157, 212,240, 244, 296, 297, 400, 407, 408
zeta function, 111critical strip, 15, 18, 29, 30, 49, 119,
132, 136, 304, 426L-function, 126η(s), 49ξ(s), 19ζ(s), 18, 19, 24, 30, 41, 297, 304, 378,
426ζK(s), 408Davenport–Heilbronn zeta function,
136Epstein zeta function, 136values of µ(σ), 142zeros, 126, 136, 297, 304, 378, 408,
426
cryptography, 61, 469, 470curve, 101, 111, 315, see elliptic curve=(ζ(s)) = 0, 118<(ζ(s)) = 0, 118algebraic, 314conductor, 124nonsingular, 100projective, 100
cusp form, 108pseudo, 123
cycle, 53graph, 53
Riemann hypothesis, 53cyclotomic polynomial, 476
finite field, 476irreducible factor, 476
Daboussi, 433, 434, 438, 439Dani, 52Dani’s theorem, 52Davenport, 136, 462Davenport–Heilbronn zeta function,
131, 134, 136de Branges, 71, 112de Bruijn, 51, 121de Bruijn–Newman constant, 51
Riemann hypothesis, 51de la Vallee Poussin, 9, 16, 47, 61, 110,
112, 119, 199, 221–224, 227, 229,268, 279, 280, 300, 304, 305, 420,425
Dedekind, 124Dedekind zeta function, 58, 408, see
zeta functions, ζK(s)ζ(s), 58
degree, 17, 108Z(t, C), 101ζK(s), 315algebraic extension, 315correspondence, 314, 315divisor, 101polynomial, 473
Delange, 434Deligne, 101, 103, 125, 126, 128, 313Deligne’s theorem, 101density hypothesis, 138Deshouillers, 62determinant
Redheffer matrix, 52
508 Index
Deuring, 127Diamond, 99differenceN(T )−N0(T ), 144, 157N(σ, T + 1)−N(σ, T ), 65NM (T )−N(T ), 157S(T )− SM (T ), 148Π(x)− π(x), 381π(x) − Li(x), 43, 134, 198, 284, 375,
376sign, 43, 134, 198, 375, 376, 381,
386, 389, 397imaginary parts of zeros, 405, 406
distribution, 406differential equation, 128Diophantine approximation, 400Dirac’s function, 406, 409Dirichlet, 107, 183, 190, 204, 206, 207,
210, 220, 225Dirichlet L-function, 56, 100, see
L-functionsDirichlet L-series, 56, 60, 64L(1, χD), 64L(s, χ), 350L(s, χk), 56bound, 64
Dirichlet character, 64, 98, 107, 124χ, 107χD, 64nonprincipal, 64primitive, 99
Dirichlet convolution, 434Dirichlet eta function, 49, 55, 56, see
alternating zeta function, see zetafunctions, η(s)
convergence, 56Riemann hypothesis, 49
Dirichlet polynomial, 121, 317, 318, 414approximative, 317, 318
Dirichlet series, 10, 11, 111, 132, 203,204, 206, 210, 219, 298, 305, 307,309, 328, 424, 426
(1− 21−2s)ζ(2s), 298summability, 298
L-functionprimitive waveform, 99
L(s,∆), 59, 108L(s, χ), 107L(s, χ3), 123
L(s, χk), 56Lν(s), 208
conjugate, 208identity with conjugate, 209
Lp(s), 57η(s), 56, 96, 325
1ζ(s)
, 49, 119, 127, 132, 285, 286, 307,308, 311, 422
convergence, 49, 285, 288derivative, 285
ζ′(s)ζ(s)
, 304, 411, 428ζ(2s)ζ(s)
, 400
log(ζ(s)), 304ψ(x), 119∑
nµ(n)
n, 307, 308, 311
ζ(s), 10, 12, 55, 95, 107, 117, 131,190, 192, 228, 302, 319, 425
convergence, 10, 55, 117, 190real line, 10
coefficient, 420cn, 27
general, 203, 204ordinary, 204Parseval’s identity, 413partial sum, 328summability, 298zero-free region, 210
Dirichlet’s class number formula, 127Dirichlet’s theorem, 356, 365, 387, 397,
400proof, 365
discrete measure, 54Riemann hypothesis, 54
discrete spectrum, 139discriminant, 108, 127distribution des nombres premiers, 211distribution des zerosLν(s), 209, 210
distribution of prime numbers, 63,95–98, 111, 118, 121, 132, 211,212, 218, 220, 222, 223, 225, 228,300–302, 354, 439, 440, 450, 469,see prime number theorem
distribution of valuesζ( 1
2+ it), 126
distribution of zerosL-functions, 111, 112, 126
upper bound, 111
Index 509
Lν(s), 209, 210MZ,f (t), 144, 147, 154Z(t), 143, 144, 146ξ(t), 193ζ(s), 41, 65, 95, 99, 118, 119, 125,
127, 138, 157, 260, 303, 319, 407,412, 459
approximation, 407upper bound, 412
zeta function, 125finite field, 125
divisor, 48, 101, 314, 315degree, 101effective, 101norm, 101positive, 101prime, 102
divisor function, 47, 48, 140divisor group, 100, 101, 103divisor problem, 66, 141
ternary additive, 141Doetsch, 426duality
prime numbers and zeros, 122Dyson, 40, 42, 100, 124, 405, 409
Edwards, 95, 136, 426Effinger, 62eigenfunction, 99eigenvalue
Frobenius endomorphism, 101, 111Gaussian unitary ensemble, 42
normalized, 42Hecke operator, 99Hermitian matrix, 124, 125, 405, 409Hermitian operator, 40, 121Laplace operator, 122linear operator, 405, 409
self-adjoint, 103zeros of ζ(s), 405, 409
linear transformation, 111matrix, 40
distribution, 103, 111orthogonal matrix, 103pair correlation, 405pair correlation function, 405, 409pseudo cusp form, 123spacing
Gaussian unitary ensemble, 42
symmetric matrix, 405, 409symplectic matrix, 103, 405, 409unitary matrix, 103, 123, 405, 409zerosΛ(s, χ), 112ζ(s), 40, 41, 121, 123, 125zeta function, 111
Eilertsen, 351Einstein, 70Einstein’s relativity theory, 70Eisenstein, 112, 183Eisenstein series, 112
inner product formula, 112Elliott, 440, 445elliptic curve, 98, 100, 110, 124, 128,
129, 471L-function, 98conductor, 110isogenous, 110modularity, 98primality test, 471rational point, 129reduction modulo p, 110
elliptic function, 298embedding
projective, 103Enestrom–Kakeya theorem, 328Enke, 46Epstein zeta function, 136equidistribution rate, 110equivalence class, 127equivalent extended Riemann hypothe-
sis, 57equivalent form, 53
Nyman–Beurling, 53equivalent functions, 330Eratosthenes, 470Erdos, 61, 317, 327, 352–354, 363, 364,
372, 420, 425, 433, 434, 439, 441Erdos’s result, 354–356ergodic measure, 52ergodic theory, 52
Riemann hypothesis, 52ERH, see extended Riemann hypothesiseRH, see extended Riemann hypothesisestimateE1(T ), 138, 139E2(T ), 138Ek(T ), 140
510 Index
N(T ), 31, 378N(T )−N0(T ), 144R2v, 31S(T ), 142Z(t), 143ζ( 1
2+ it), 138
ζ(s)at infinity, 420
coefficient, 348Euclid, 301, 302Euclid’s theorem, 301, 302Euler, 10, 29, 62, 96, 118, 190, 230, 285,
301, 302, 319Euler product, 10, 11L-function
global, 98primitive waveform, 99
L(s, χ), 107L(s, χ3), 124ζ(s), 11, 16, 96, 107, 118, 131, 136,
190, 228, 302, 304, 319, 425, 427,428
convergence, 190logarithm, 16, 96logarithmic derivative, 229, 304,
427, 428ζ(s, C), 101Z(s), 137Davenport–Heilbronn zeta function
lack, 136fundamental theorem of arithmetic,
11zeta function
number field, 124Euler product formula, see Euler
productEuler’s constant, 48, 52, 135, 230, 434
Riemann hypothesis, 52Euler’s factorisation formula, 422Euler’s phi function, 127, see Euler’s
totient functionEuler’s totient function, 48, 54, 58, 473,
see Euler’s phi functionEuler–MacLaurin summation formula,
29, 38, see algorithmsimprovement, 32Riemann–Siegel formula, 32
evaluation∆, 272
∑ρ
1σ2+t2
, 233∑ρ
yσ−1
σ2+t2, 264∑
ρyσ−1
t2, 260∑
pmlog(p)
pm , 276∑pm log(p), 268
explicit formulaψ(x), 119, 462Motohashi, 139Riemann, 102, 119Weil, 102, 122, 123L-function, 102ζ(s), 102zeta function, 102
exponential decay, 13, 144exponential time, 471extended Riemann hypothesis, 57, 58,
469arithmetic progression, 63consequence, 61, 63, 469, 471equivalent statement, 57, 58generalized Riemann hypothesis, 57primality test, 469, 471
Farey, 48Farey series, 48Fn, 48Haros, 48Riemann hypothesis, 49term, 48
Farmer, 126fast Fourier transform algorithm, see
algorithmsFejer, 317, 327Fermat, 301, 470Fermat’s last theorem, 4, 99, 107, 120Fermat’s little theorem, 110, 301, 470,
471extension, 301finite field
primality test, 471generalization, 471, 472polynomial ring
finite field, 471primality test, 470, 471
FFT, see fast Fourier transformfield, see finite field, see function field,
see number fieldalgebraic closure, 315
Index 511
algebraically closed, 103Galois, 315imaginary quadratic, 127
Fields medal, 27, 313finite field, 100, 101, 111, 125, 314, 471,
473, 476Fp, 473, 476Fq, 100, 101polynomial ring, 471
fixed point formula, 102flow, 52
horocyclic, 52fonction
elliptique, 298Mobius, 434Riemann, 200, 222–224, 228Schlomilch, 204Von Mangoldt, 434
form factor, 405, 409forme quadratique, 220formule
Cahen, 297Gauss, 230, 237, 243Legendre, 173, 175, 179Riemann, 297
Fourier analysis, 420Fourier integral, 426Fourier inversion, 97Fourier transform, 51, 121, 122, 144,
407, 420, 452, 463of Ξ(t), 121
Fourier’s theorem, 194Fouvry, 479fractional part, 12, 30, 52, 121, see
functions, xFrey, 107Friedlander, 128Frobenius endomorphism, 101, 103
cohomology group, 111eigenvalue, 101, 111fixed point, 102Lefschetz fixed point formula, 102trace, 102
Fujii, 147, 459function field, 313, 314
Riemann hypothesis, 313, 314functional analysis, 121, 199functional equationL-function
elliptic curve, 98global, 98primitive waveform, 99
L(s,∆), 59, 108L(s, χ), 108L(s, χ3), 124LE(s), 124L∆(s), 124Lν(s), 209Λ(s, π), 59, 109µ(σ), 141ϑ(t), 13, 25ξ(s), 14, 20ζ(s), 14, 15, 95–97, 107, 117, 131,
191, 192, 280, 304, 319logarithmic derivative, 411symmetric, 117, 150
ζ(s, C), 101ζK(s), 316Z(s), 137alternating zeta function, 96approximate, 32, 33, 120Davenport–Heilbronn zeta function,
136zeta function
number field, 124functions, see L-functions, see arith-
metic functions, see convolutionfunctions, see Dirichlet series,see mean, see partial sum, seeSchlomilch function, see zetafunctions
A(u), 399, 400A∗T (u), 399, 400Bn(x), 30Ek(T ), 137Em, 33F (X,T ), 447, 449, 459F (α), 405, 406F (α, T ), 405, 406F (x), 193, 227, 278, see functions,
π(x)F (y, λ), 390, 394G, 149, see Barnes’s G functionH(λ, z), 51H(z), 26, 27I, 145, 155I1, 145, 152I1(T ), 67
512 Index
I2, 145, 152I2(T ), 67Ik(T ), 66, 67, 125J(X,T ), 463Jµ(x), 212L(s,∆), 59L(x), 332, 399, 400L1(x), 342Lp(s), 57M(x), 47, 69, 119, 307–309, 402,
433–435, 445, see Mertens’function
N(T ), 9, 15, 19–21, 24, 31, 35, 38, 41,99, 119, 132, 378, 459
N(α, T ), 100N(σ, T ), 65, 138N0(T ), 119, 144N1(T ), 38NM (T ), 144R(T ), 378R(t, x), 151R(x), 359, 360S(N), 35S(T ), 119, 132S(t), 112S(t, x), 151SM (T ), 147U , 155Z(t), 32, 33, 35, 133–135, 142, 143,
150, 151even, 133logarithmic derivative, 135real, 133
Z(k)(t), 143Z2(ξ), 139[x, x1, x1, . . . , xn], 146[x], 12, 428∆, 20, 122, 270∆(t), 150∆(x), 140∆(z), 59, 108, 124∆(k)(t), 150∆2(x), 140∆k(x), 140Γ (s), 12, 13, 15, 22, 26, 107, 118Λ(s), 107Λ(s,∆), 108Φ(t), 51, 121Π(s− 1), 190
Π(x), 97, 377, 381Π0(x), 377, 378, 388
<( Γ ′(a)Γ (a)
), 233
<( ζ′(s)ζ(s)
), 22
<(log(ζ(s))), 16Ξ(iz), 51Ξ(t), 121, 297χ(s), 32, 131η(s), 49
1ζ(s)
, 49, 119, 132, 285derivative, 285
Γ ′(a)Γ (a)
, 231, 237Γ ′(s)Γ (s)
, 22log(ζ(s))
s, 194
ξ′(s)ξ(s)
, 50Riemann hypothesis, 50
ξ(s)ξ(s)
, 21, 22ζ′(s)ζ(s)
, 22, 23, 214, 216, 229, 245, 304s = 0, 268, 279
ζ′(s)ζ2(s)
, 285
f(x), see Fourier transformln(x), 473log((s− 1)ζ(s)), 96log(Γ (a)), 231log(Γ (s)), 21log(ξ(t)), 193log(ζ(s)), 16, 97, 193, 200, 201, 203,
304real part, 201singularity, 97
log(s− 1), 211log(x), 433, 473S(T ), 21, 24, 65, 66S1(T ), 65, 66µ(σ), 140φ, 101φ(t), 121π(x), 7, 18, 27, 43, 46, 61, 96, 97, 118,
119, 132, 161, 162, 222, 227, 278,302, 365, 375–377, 420, 425, 447,see functions, ϕ(x)
π(x)− Li(x), 134, 375, 376π(x; k, l), 57, 58, 63, 127π2(x, k), 458ψ(x), 96, 119, 305, 307–309, 377, 379,
427, 429, 448
Index 513
ψ0(x), 377, 379ψ1(x), 380ψµ(x), 214σ(y), 267, 268, 271, 273, 275, 283sin θ(t), 33θ(t), 32, 33, 150θ(x), 303ϕ(u, t), 248ϕ(x), 163, 207, see functions, φ(x)ϑ(t), 13, 25, 26, see Jacobi theta
functionϑ(x), 96, 354, 365ξ( 1
2+ it), 297
ξ(t), 38ξ(s), 14, 19–22, 25–27, 31, 38, 50, 95,
118, 135, 297ξ(s, χ3), 124ξ(t), 97, 192, 193ξE(s), 124ξ∆, 124ξk(s), 204, 205ζ′(s), 49
convergence, 49ζ( 1
2+ it), 100, 117, 120, 126, 131
x, 121f(x), 173, 194k, 147, 156k-tuple correlation, 409k(t,H), 147k(t, T ), 156Li(x), 18, 46, 58, 96, 132, 162, 197,
222, 225, 281, 375, 376, 388x, 12, 30, 52pair correlation, 405, 409
fundamental theorem of arithmetic, 10,118
analytic form, 11Euler product, 11
Gallagher, 449, 451, 456Galois field, 315gamma function, 12, 13, 15, 22, 26, 107,
118logarithmic derivative, 22
Gauss, 43, 46, 96, 118, 127, 134, 183,190, 198, 225, 230, 237, 243, 301,303, 304, 425
Gauss measure, 124Gauss sum, 109
sign, 108Gauss’s class number problem, 127Gauss’s conjecture, 118, 119
proof, 127restatement, 119
Gauss’s formula, 230, 237, 243Gaussian unitary ensemble, 40, 42, 111,
124eigenvalue, 42
normalized, 42eigenvalue spacing, 42
Gaussian unitary ensemble conjecture,125
Gel’fand, 143Gelfond, 120, 147general divisor function, 140, see
arithmetic functions, dk(n)general linear group, 58, 108generalized Riemann hypothesis, 56
Artin’s primitive root conjecture, 64,479
consequence, 62, 64extended Riemann hypothesis, 57Goldbach’s conjecture, 62Miller–Rabin primality test, 63primality test, 63Riemann hypothesis, 57Solovay–Strassen primality test, 63
generic point, 314genus, 101, 315geodesic, 123geodesic motion, 110
surface, 110Germain, 471, 479Ghosh, 67, 125Gilbert, 222GLH, see grand Lindelof hypothesisGoldbach, 62, 301Goldbach conjecture, 62, 301
strong, 62weak, 62
Goldbach type, 110Goldfeld, 128, 479Goldschmidt, 198Goldston, 447, 448, 450, 459Goldwasser, 471Goldwasser–Kilian primality testing
algorithm, 471Gonek, 125
514 Index
Goursat, 199Gram, 29, 33, 34, 285Gram block, 34
consecutive, 34Gram point, 33–35
bad, 34good, 34
Gram’s law, 33–35, 410exception, 34
grand Lindelof hypothesis, 109approximation, 110consequence, 109
grand Riemann hypothesis, 58–60, 106,107, 109
Λ(s, π), 113approximation, 110, 111consequence, 107, 109, 415Dirichlet L-functions, 110
consequence, 110verification, 112
graph, 53adjacency matrix, 53cycle, 53of Z(t), 142
graph theory, 53GRH, see grand Riemann hypothesisgRH, see generalized Riemann
hypothesisGronwall, 338Gronwall’s inequality, 338Gross, 128Grothendieck, 101groupG, 476
cardinality, 476, 477Sn, 53Γ0(N), 98, 99GLm, 108GLm(A), 58, 108GLm(Qv), 108SL(2,Z), 122G, 476
cardinality, 476, 477Div(C), 100cohomology, 111conjugacy class, 137divisor, 100, 101, 103element, 53
order, 53
general linear, 58, 108Hecke, 98ideles, 112modular, 108, 139monodromy, 111rational points
elliptic curve, 129semi-simple, 112symmetric, 53
GUE, see Gaussian unitary ensembleGUE conjecture, 125Gupta, 64, 351
Haar measure, 125, 126Haas, 123Hadamard, 15, 47, 61, 110, 119, 199,
200, 205, 215, 222, 224, 227, 300,304, 305, 313, 420, 425
Hadamard product representation, 21logarithmic derivative, 21
Hadamard–de la Vallee Poussintheorem, 199, 222
Hafner, 138Halasz, 446Halberstam, 458Halphen, 212, 218, 219Halphen’s theorem, 212, 218, 219Hamiltonian, 110
eigenstate, 110Hardy, 9, 24, 31, 32, 51, 62, 67, 96, 110,
120, 123, 125, 138, 296, 297, 300,301, 307, 308, 456
Hardy and LittlewoodRiemann hypothesis, 51
Hardy’s Z-function, 32, 33, 35Hardy’s function, 31Hardy’s theorem, 24, 27
proof, 27Riemann hypothesis, 24, 27
harmonic form, 103harmonic number, 48harmonic series, 10, 11, 16Haros, 48
Farey series, 48Haselgrove, 399–401Hasse, 101Hasse–Weil L-function, 124, 129Heath-Brown, 63, 64, 72, 100, 450Heath-Brown’s theorem, 64
Index 515
Artin’s primitive root conjecture, 64Hecke, 124, 127Hecke group, 98Hecke operator, 99
eigenvalue, 99Heilbronn, 127, 136Heins, 420, 426Hejhal, 121, 123, 136Hermite, 199Hermitian linear operator, 405, 409Hermitian matrix, 124, 125, 405, 409,
410eigenvalue, 124, 125, 405, 409
Hermitian operator, 40, 121eigenvalue, 40, 121
heuristics, 130Hilbert, 4, 37, 40, 52, 103, 111, 117,
120, 121, 124Hilbert space, 112L-functions, 112
Hilbert–Polya conjecture, 40, seeconjectures
Hilbert–Schmidt integral operator, 52Riemann hypothesis, 52
Hildebrand, 438, 439Hodge, 103Hodge’s theorem, 103Hooley, 64, 98horocyclic flow, 52Huang, 471Hudson, 44Hurwitz, 136, 204, 206, 209, 325Hurwitz zeta function, 136, see zeta
functions, ζ(s, a)Hurwitz’s theorem, 325Hutchinson, 29, 34Huxley, 138hyperbolic Laplacian, 99hypothesis H, 44, 375, 377, 380
consequence, 375, 377, 378, 380–382,384, 386, 388, 395
Riemann hypothesis, 44
ideal, 58ideal class, 127idele class group, 112Ikehara, 420, 424, 426Ikehara–Wiener Tauberian theorem,
424, 426
“poor man’s” version, 424, 426prime number theorem, 424, 426
imaginary quadratic field, 127inegalite de Selberg, 434infimum
1− σn, 227, 247, 251, 256
M(n)n−12 , 70
upper bound, 70M(x)
x, 434
zerosimaginary parts, 408, 410
Ingham, 67, 120, 125, 296, 376, 383,399, 400, 402, 419, 420, 426
Ingham’s convergence theorem, 420proof, 420–422
Ingham’s exponent, 100Ingham’s theorem, 419
proofNewman, 419
integerintrospective, 475, 476
closed under multiplication, 475,476
residue, 476set, 476
modulo n, 473, 476residue, 476
odd, 51square, 64sum
prime numbers, 62integer ring, 473OK , 58
integral ideal, 58norm, 58
integral operator, 52integral part, 12, see functions, [x]integral representationF (X,T ), 459MZ,f (t), 143
M(k)Z,f (t), 143
N(T ), 99Z2(ξ), 139[x, x1, x1, . . . , xn], 146Γ (s), 13Π(s− 1)ζ(s), 191Γ ′(a)Γ (a)
, 231, 237log(ζ(s))
s, 194
516 Index
ζ′(s)ζ(s)
, 429
log(Γ (a)), 231ψ(x), 305ξ(t), 192ξk(s), 205ζ(s), 12, 96, 97, 428Li(x), 46, 58, 132, 225, 281, 376
intermediate value theorem, 38internet security, 469intersection number, 314, 315introspective, 475, 476, see integer
polynomial, 475inverse Fourier transform, 149inversion, 197inversion formula, 97irreducible correspondence, 314irreducible polynomial, 473, 476irreducible representation
unitary, 58, 108isogenous, 110Ivic, 100, 130, 131, 138–140, 143Iwaniec, 98, 113, 116, 127, 128
Jacobi, 183, 192, 314, 471Jacobi symbol, 471, see arithmetic
functions,(
an
)primality test, 471
Jacobi theta function, 13, 25, 26, seefunctions, ϑ(t)
Jacobi’s inversion theorem, 314Jensen mean, 329, 352, see meanJessen, 323, 350Jutila, 142
Kalai, 477Karatsuba, 136Katz, 103, 125, 126Kayal, 63, 469, 470Keating, 67, 111, 125, 126Keating–Snaith conjecture, 111, 125,
126Kilian, 471Klein, 133Knopp, 322Korevaar, 61, 424, 425Korobov, 18Kossuth Prize, 317Kowalski, 113Kristensen, 351
Kubilius, 440, 445, 446Kuznetsov, 141
Lacroix, 238Lagarias, 48, 50, 123Landau, 6, 7, 95, 100, 127, 139, 296,
297, 301, 329, 333, 341, 419, 420,422, 426
Landau’s classical result, 139Landau’s paradigma, 341Landau’s theorem, 333, 334, 341Landau–Siegel zero, 127, 128Langlands, 59, 109, 112Langlands conjecture, 483Langlands program, 124Laplace, 122, 424, 426, 427Laplace integral, 424, 426Laplace operator, 122
eigenvalue, 122Laplace transform, 426, 427large sieve, 438, 439
exponential sum, 445Montgomery, 440prime number theorem, 438, 439standard version, 445
large sieve inequality, 438, 440, 445, 446Elliott, 440, 445prime number theorem, 438proof, 440, 445Turan–Kubilius inequality, 440, 445,
446Laurent series, 17Lavrik, 143lcm, see least common denominatorLCM(m), see least common denomina-
torleast common denominator, 473Lebesgue, 383, 431Lebesgue integral, 383Leech, 401Lefschetz fixed point formula, 102Legendre, 118, 163, 173, 175, 179, 182,
238, 303, 304, 425Legendre symbol, 57, see arithmetic
functions,(
np
)Legendre’s conjecture, see conjecturesLegendre’s formula, 173, 175, 179Lehman, 34, 44, 134Lehmer, 43, 401
Index 517
Lehmer’s phenomenon, 43, 131, 134,136
Lejeune, 225Lenstra, 476, 480Levinson, 27, 100, 120LH, see Lindelof hypothesisLi, 71limite inferieure
1− σn, 227, 247, 251, 256limite superieureσn, 227, 244, 247
Lindelof, 65Lindelof hypothesis, 65, 120, 137, 138
conditional disproof, 141equivalent statement, 65, 138, 140,
141grand, see grand Lindelof hypothesisimplication, 138Riemann hypothesis, 65, 120, 137,
138statement affecting, 140statement implying, 65zeros of ζ(s), 138
line of symmetry, see critical linelinear operator, 103, 405, 409
eigenvalue, 103, 405, 409Hermitian, 405, 409self-adjoint, 103unitary, 405, 409zeros of ζ(s), 405, 409
linear transformationeigenvalue, 111
Liouville’s function, 6, 8, 46, 330, 399,400, see arithmetic functions, λ(n)
prime number theorem, 7Riemann hypothesis, 6, 46
Lipschitz, 209Littlewood, 43, 51, 62, 64, 67, 98, 110,
120, 123, 125, 134, 138, 296, 307,308, 375–378, 390
Littlewood’s theorem, 376local representation, 58, 108location of prime numbers, 63location of zeros
(1− 21−2s)ζ(2s), 298Jn(s), 329L-functions, 98L(s, χ), 350Lν(s), 209
MZ,f (t), 144Un(s), 322, 335Vn(s), 343, 349Z(t), 144Λ(s), 107Λ(s, χ), 112Λ(s, π), 111ξ(s), 38ξ(t), 193ζ(s), 14, 15, 24, 30, 70, 95, 99, 100,
112, 119, 120, 131, 132, 191, 199,200, 228, 240, 244, 260, 296, 297,304, 319, 320, 375–378, 380, 399,401, 403, 404, 412
Λ(s, π), 109Davenport–Heilbronn zeta function,
136partial sum, 71, 317
locus, 314lois asymptotiques relatives aux nombre
premiers, 260lower bound
1− σ, 227, 247, 251, 256, 294, 295E1(T ), 138E2(T ), 138F (X,T ), 458L(1, χk), 127L(n), 332S(t), 112Z(t), 143Λ, 51γn, 229π(x), 96, 97, 303ψ(x), 97∑
ρ1
s−ρ, 243∑
plog(p)
p, 372, 374
ϕ(x), 168ϑ(x), 97, 355h(d), 127, 128on(a), 474cardinality of G, 476, 480class number, 128Dirichlet polynomial, 121LCM(m), 473product of prime numbers, 480supremum
M(n)n−12 , 70
zeros
518 Index
ζ(s), 136, 417Davenport–Heilbronn zeta function,
136imaginary parts, 294, 416
Lucas, 301
Mobius function, 47, 69, 119, 127, 128,132, 284, 356, 402, 419, 433, 434,438, 439, see arithmetic functions,µ(n)
Maass L-function, 100Maass waveform, 99, 104Maass–Selberg formula, 112MacLaurin, 29MacLaurin summation formula, 99Mansion, 224, 225, 295matrix
adjacency, 53eigenvalue, 40, 103, 111Hermitian, 124, 125, 405, 409, 410orthogonal, 103, 126random, 40, 42, 106Redheffer, 52
determinant, 52symmetric, 405, 409symplectic, 103, 126, 405, 409unitary, 67, 103, 123, 405, 409unitary symplectic, 126
maximumZ(t), 134–136ξ(t), 40, 100ζ( 1
2+ it), 142
mean, 328Cn(s), 324, 326E1(T ), 140E2(T ), 140Jn(s), 329Rn(s), 324, 326analytical, 356arithmetical, 328Cesaro, 298, 324, 326, see mean,
Cn(s)Jensen, 329, 352, see mean, Jn(s)Riesz, 324, 326, see mean, Rn(s)
mean squareS(t), 113
asymptotic behavior, 113ζ( 1
2+ it), 156
mean value
MZ,f (t), 144, 155Z(t), 144ζ( 1
2+ it), 141
asymptotic behavior, 141ζ(s), 66
mean value formula, 134, 137critical line, 137
mean value theorem, 17, 146Wirsing, 439
measure, 52, 54discrete, 54ergodic, 52
Mehta, 42Mellin, 297Mellin inversion formula, 25Mellin transform, 102, 426, 427meromorphic function, 19, 59, 139Mertens, 37, 69, 133, 284, 285, 402, 428,
439, 441Mertens’ conjecture, 69, 119, 133, 402,
404conditional disproof, 402disproof, 69, 119
Odlyzko-te Riele, 69Riemann hypothesis, 69
Mertens’ formula, 441Mertens’ function, 47, 48, 69, 402, 438,
see functions, M(x)order, 69, 70Riemann hypothesis, 47
Mertens’ theory, 439Meyer, 220Meyer series, 220Millennium Prize, 4, 72, 94, 107, 113Miller, 63, 110, 401, 471Miller’s primality testing algorithm, 98,
110, 471running time, 110
minimumZ(t), 134–136ξ(t), 40, 100ζ( 1
2+ it), 142
modular form, 107, 124modular function, 98modular group, 108, 139module, 315
correspondence, 315modulus, 64moment, 128, 129
Index 519
L( 12
+ it, π), 111asymptotics, 111
L( 12, π), 111
asymptotics, 111ζ( 1
2+ it), 120, 131, 137, 140, 141
ζ(s), 66, 125Brownian bridge, 121characteristic polynomial, 125, 126family of L-functions, 126zeta function, 130
monodromy group, 111Montgomery, 40–42, 71, 100, 103, 124,
125, 128, 142, 317, 405, 406, 440,447–449, 459, 463
Montgomery’s conjecture, see conjec-tures
Montgomery’s large sieve, 440Montgomery’s pair correlation
conjecture, 41, 43, see conjecturesRiemann hypothesis, 41
Montgomery–Odlyzko law, 42, 111Motohashi, 127, 138–141Motohashi’s fundamental explicit
formula, 139moyenne de Cesaro, 298Mueller, 449, 450, 459multiple poleZ2(ξ), 139
multiple zeroL-function, 128, 129
elliptic curve, 129triple, 128
ξ(t)odd, 38
multiplicity, see orderpk, 147poleL1(s), 208Z(t, C), 101Z2(ξ), 139ξk(s), 205ζ(s), 11, 15, 95, 99, 117, 131, 200,
304, 319ζ(s, C), 101
zeroL-function, 128MZ,f (t), 144Z(t), 133, 143, 144ξ(t), 38
ξ(t), 99ζ(s), 39, 100, 122, 128, 133, 134,
400, 407–409, 416elliptic curve L-function, 129
Newman, 51, 61, 121, 419, 420, 424–427non-Euclidean Laplacian, 139
discrete spectrum, 139nonsingular model, 315nontrivial zero, 5, 15, 100, 227L(s, χ3), 124L(s, χk), 57L∆(s), 124Lν(s), 209Λ(s), 107η(s), 49ξ( 1
2+ it), 121
ζ(s), 5, 15, 18, 38–41, 70, 95, 97, 119,131–135, 157, 228–230, 233, 260,296, 297, 303, 304, 319, 326, 375,376, 399, 400, 405, 406, 426, 447,449, 456
lim inf(1− σn), 227, 247, 251, 256lim sup(σn), 227, 244, 247ρ, 119, 135, 228ρn, 399, 400argument principle, 19conjugate, 229counting, 18, 19distribution, 407existence, 27, 38, 42, 304, 407proportion, 27simple, 100smallest, 35, 38, 41, 70, 99, 100,
131, 229, 376, 387, 388, 399–401symmetric, 15
ζK(s), 58Davenport–Heilbronn zeta function,
136norm
conjugacy class, 137divisor, 101integral ideal, 58
NP, see complexity classnumber field, 58number of primes, 95, 183, 190, 198numeratorZ(t, C), 101ζK(s), 315
520 Index
degree, 315polynomial, 315
Nyman, 53, 121Nyman–Beurling equivalent form, 53
Riemann hypothesis, 53Nyman–Beurling theorem, 121
Odlyzko, 35, 42, 51, 69, 70, 100, 103,111, 112, 119, 120, 125, 133, 134
Odlyzko–Schonhage algorithm, 35operator, 26
Hecke, 99Hermitian, 40, 121integral, 52
Hilbert–Schmidt, 52Laplace, 122linear, 103, 405, 409, see linear
operatorself-adjoint, 111
orbit, 52closed, 52
order, see multiplicity(s− 1)ζ(s), 137E1(T ), 137E2(T ), 137, 139Ek(T ), 140Em, 33M(x), 69, 70, 119
M(k)Z,f (t), 144
N(T )−N0(T ), 157N(α, T ), 100
Ingham’s exponent, 100NM (T )−N(T ), 157R(x), 361S(T ), 119, 132Z(t), 143Z(k)(t), 143Π(x)− π(x), 381S(T ), 66S1(T ), 66π(x), 302, 303π(x)− Li(x), 227, 284ψ(x), 305σ(y), 267, 268∑
ρyσ−1
σ2+t2, 265–267∑
ρyσ−1
t2, 263, 264∑
nµ(n)
n, 288
θ(x), 303
ϑ(x), 366ξ(s), 135ζ( 1
2+ it), 120, 138
ζ(s)bound, 138critical line, 138Z(s), 137Li(x), 380group element, 53integer modulo n, 473symmetric group element, 53
equivalent statement, 54orthogonal matrix, 103, 126
eigenvalue, 103oscillations
of Z(t), 142, 143
P, see complexity classPolya, 37, 40, 103, 111, 121, 124, 129,
296, 333, 335, 351, 400Polya’s conjecture
Riemann hypothesis, 399–401Polya’s theorem, 333, 334poleL1(s), 208Lν(s), 208Γ (s), 228ξk(s), 205ζ(s), 200, 203
simple, 200pair correlation, 124, 128, 405, 406, 409,
447–449, see spacing betweeneigenvalue, 405
symmetric matrix, 405, 409symplectic matrix, 405, 409
Montgomery, 463prime numbers, 447, 448, 450short interval, 447, 448zeros of ζ(s), 42, 409, 447, 448, 459
pair correlation conjecture, 449pair correlation function, 42, 405, 409
eigenvalueGaussian unitary ensemble, 42Hermitian matrix, 405, 409unitary matrix, 405, 409
plot, 42Parseval, 413Parseval’s identity, 413partial sum, 317, 350
Index 521
L(s, χ), 350Un(s), 320, 324, 326Vn(s), 325, 343Wn(s), 330rn(s), 322alternating zeta function, 325, see
partial sum, Vn(s)Dirichlet series, 328power series, 352Riemann hypothesis, 71, 317, 320,
321, 324, 325Riemann zeta function, 71, 317, 320,
see partial sum, Un(s)zero, 317, 327, 328
behavior, 322condensation point, 327, 328, 352
zero-free region, 317paths
0 1, 1510 1, 151
peak, 154Z(t), 142, 143ζ( 1
2+ it), 100
ζ(s), 40perfect square, 479Petersen, 351Phragmen–Lindelof theorem, 120Piltz, 210Plancherel, 463Plancherel’s identity, 463Poincare, 211Poisson summation, 13poleL-function, 98
location, 98L1(s), 208
multiplicity, 208simple, 208
Lν(s), 208Z(t, C), 101
multiplicity, 101simple, 101
Z2(ξ), 139critical line, 139multiplicity, 139multiplicity five, 139simple, 139
Γ (s), 13, 14, 228residue, 13
simple, 14ζ′(s)ζ(s)
residue, 229, 230ξk(s), 205
multiplicity, 205residue, 205simple, 205
ζ(σ), 17residue, 17simple, 17
ζ(s), 11, 14, 15, 38, 95, 99, 117, 127,131, 191, 200, 203, 304, 319
multiplicity, 11, 14residue, 11, 15simple, 11, 14, 15, 95, 99, 117, 131,
200, 304, 319ζ(s, C), 101
multiplicity, 101simple, 101
meromorphic function, 19counting, 19
residue, 38zeta functions, 101
polynomialQn(X), 476ζK(s), 315Artin L-function, 314Bernouilli, 30characteristic, 67, 125Chebyshev, 36cyclotomic, 476degree, 473introspective, 475irreducible, 473, 476residue, 476
polynomial ring, 471Fermat’s little theorem, 471finite field, 471
polynomial time, 469–471Pomerance, 471, 480positive integer
introspective, 475, 476perfect square, 479
Poulsen, 351power series, 352
partial sum, 352Pratt, 471premiers de la progression arithmetique,
227
522 Index
primality testing algorithms, 63, 301,469, 470, 472, see Agrawal–Kayal–Saxena primality test
Adleman–Huang, 471Adleman–Pomerance–Rumely, 471Atkin, 471conditional, 471elliptic curve, 471exponential time, 471
deterministic, 471extended Riemann hypothesis, 469,
471Fermat’s little theorem, 470
polynomial ring, 471generalized Riemann hypothesis, 63Goldwasser–Kilian, 471Jacobi symbol, 471Miller, 110, 471
running time, 110Miller–Rabin, 63
conditional deterministic, 63conditional polynomial time, 63generalized Riemann hypothesis, 63
polynomial time, 469–472deterministic, 469–471proof, 469, 472randomized, 471, 472
Rabin, 471reciprocity law, 471running time, 470, 472Solovay’s primality test, 471Solovay–Strassen, 63
conditional deterministic, 63generalized Riemann hypothesis, 63
Sophie Germain prime density, 471,479
time complexity, 471, 479Strassen, 471unconditional, 469–471
primality testing problem, 471prime, see prime number
divisor, 102representation, 58
prime k-tuple hypothesis, 451weak, 451
prime counting function, 46, 183, 190,198, 227, 278, 425
see functions, π(x), 483prime decomposition, 301
prime number, 63, 64, 300, 301, 469,470
accumulation, 190arithmetic progression, 63, 107, 127,
220, 227, 458distribution bound, 111upper bound, 458
asymptotic behavior, 212, 218, 220,222, 225, 226, 260, 300, 301, 425,450, 479
bad, 367cardinality, 301, 302decomposition, 301density, 198, 479
sieve, 471distribution, 18, 63, 95–98, 111, 118,
121, 132, 211, 212, 218, 220, 222,223, 225, 228, 300–302, 354, 439,440, 450, 469, see prime numbertheorem
sieve, 439upper bound, 439, 440
estimate, 439good, 367, 371
existence, 367, 369, 372interval, 63large, 470location, 63, 354pair, 41, 124, 301
distribution, 41, 124pair correlation, 447, 448, 450power, 461
location, 461product, 301, 480
lower bound, 480property, 470Riemann zeta function, 46set, 470Sophie Germain, 471, 479
density, 471, 479spacing, 450sum of three, 62sum of two, 62twin, 41, 124
constant, 479distribution, 41, 124
zeros of ζ(s), 410prime number theorem, 7, 43, 46, 61,
109, 119, 162, 172, 199, 222,
Index 523
284, 300–303, 307–309, 353, 354,363–365, 419, 420, 424, 425,433, 434, 469, see theorems,π(x) ∼ x
log(x)
arithmetic progression, 356, 365proof, 356, 365
auxiliary Tauberian theorem, 427conditional proof, 162, 172consequence, 414Dirichlet series, 307equivalent statement, 8, 303, 305,
307, 309, 354, 361, 365, 419, 420,422, 427, 429, 438, 439, 441
exposition, 300Gauss, 46history, 300, 301, 424, 426Ikehara–Wiener Tauberian theorem,
424, 426intuitive statement, 8large sieve, 438, 439large sieve inequality, 438Liouville’s function, 7proof, 18, 119, 199, 222, 285, 302,
307, 308, 353, 354, 363, 364, 374,419, 420, 434
analytic, 8, 419, 420Daboussi, 433, 438, 439de la Vallee Poussin, 8, 9, 47, 61,
119, 222, 300, 304, 305, 420, 425elementary, 8, 61, 353, 354, 363,
364, 371, 433, 438Erdos, 61, 353, 363, 364, 420, 425,
433, 439, 441Erdos-Selberg, 363, 372, 373Hadamard, 8, 47, 61, 119, 199, 222,
300, 304, 305, 420, 425Hardy–Littlewood, 307, 308Hildebrand, 438, 439Ikehara, 420Korevaar, 8, 61, 424, 427Newman, 61, 419, 420, 422, 424, 426Selberg, 61, 353, 354, 361–363, 365,
369, 420, 425, 433, 439simplification, 363, 365, 371, 372Wiener, 420
Riemann hypothesis, 43similar theorem, 307, 308Wiener theory, 426
primitive character, 56
primitive root, 64primitive waveform, 99Primzahlsatz, 301principal character, 56principal valueπ(x), 282Li(x), 226
problems of the millennium, 94, 95, 106,107, 131
product representation, see Eulerproduct
L(s,∆), 59, 108L(s, χ3), 124L(s, π), 59L(s, π∞), 59L(s, πp), 59LE(s), 124L∆(s), 124Lν(s), 208
1ζ(s)
, 119, 285
ξ(s), 15Hadamard, 21
logarithmic derivative, 21progression arithmetique
formes quadratiques, 220nombres premiers, 220, 227
projective curve, 100nonsingular, 100
projective variety, 103nonsingular, 103
pseudo cusp form, 123eigenvalue, 123
Purdue, 71
quadratic form, 220, 316arithmetic progression, 220equivalence class, 127negative semidefinite, 103self-intersection, 103ternary, 110
quadratic twist, 126quantum chaos, 110
Rabin, 63, 471Rabin’s primality testing algorithm, 471racine, see zero, see zeroRademacher, 63, 70, 120Rains, 48Ram Murty, 64
524 Index
Ramachandra, 67, 141, 142Ramanujan, 110, 296Ramanujan property, 98Ramanujan’s tau function, 124, see
arithmetic functions, τ(n)random matrix, 40, 42, 106
Riemann hypothesis, 40random matrix ensemble, 111random matrix theory, 40, 42, 116,
124–126, 128, 129, 131Hilbert, 40Polya, 40Riemann hypothesis, 40
random walk, 6, 7rational function, 35, 315Z(t, C), 101
rational point, 128, 129elliptic curve, 129
reciprocity law, 471primality test, 471
Redheffer, 53, 123Redheffer matrix, 52
determinant, 52Riemann hypothesis, 53
relationπ(x) and ψ(x), 429σ and t, 120, 240, 244, 250, 251, 289,
294, 295Renyi, 445representation, 58, 108π, 58, 59, 108π∞, 58, 108πp, 58, 108πv, 58, 108automorphic, 98automorphic cuspidal, 58, 108conductor, 109contragredient, 59, 109irreducible unitary, 58, 108local, 58, 108prime, 58
residue, 38, 97integer modulo n, 476introspective integer modulo n, 476pole, 38
ζ′(s)ζ(s)
, 229, 230
ξk(s), 205ζ(σ), 17
ζ(s), 15polynomial, 476
residue theorem, 38, 146, 147, 153, 421RH, see Riemann hypothesisRibet, 107Richert, 458Riemann, 3, 4, 9, 10, 12, 14, 15, 19, 31,
32, 38, 39, 43, 70, 95, 97, 98, 107,117–119, 132, 133, 161, 183, 190,200, 211, 226, 297, 300, 303, 304,431
Riemann hypothesis, 3–6, 15, 29, 37,55, 61, 70, 93–95, 97, 99, 106,107, 116, 117, 119, 120, 130–133,154, 157, 161, 183, 296, 304,307, 319, 376, 399, 426, seeextended Riemann hypothesis, seegeneralized Riemann hypothesis,see grand Riemann hypothesis
L-functions, 99, 104, 124consequence, 98global, 98Maass waveform, 100modular form, 100primitive waveform, 99verification, 100
L(s, χ3), 124L(s, χk)
implication, 127verification, 126
LE(s), 124L∆(s), 124Z(t, C), 101Λ(s), 107Λ(s, π), 113η(s), 56ξ′(s)ξ(s)
, 50λn, 50ζ(s, C), 101ζK(s), 58, 313, 316
consequence, 316proof, 313, 316Z(s), 137
true, 137algorithm, 29axioms, 124Bertrand’s postulate, 63conditional disproof, 130, 131, 141,
157
Index 525
connection, 137, 328
Lindelof hypothesis, 137
consequence, 41, 43, 44, 61, 63, 65–67,322, 335, 376, 399–401, 405–408,447–451, 458
Titchmarsh, 65
convolution function, 131, 154
curve, 111
proof, 125
cycle
graph, 53
Davenport–Heilbronn zeta function
false, 136
de Bruijn–Newman constant, 51
Dirichlet L-function, 99, 100
Dirichlet eta function, 49
discrete measure, 54
disproof, 93, 154, 157
false, 70
Rademacher, 70
doubts, 93, 130, 131, 134, 137, 154,157
Epstein zeta function
false, 136
equivalent statement, 6, 18, 45–54,56, 70, 98, 100, 103, 112, 119,121–123, 132, 133, 144
analytical, 49
number theoretic, 45
ergodic theory, 52
Euler’s constant, 52
evidence, 18, 32, 37, 39, 99, 102, 112,120, 134, see verification
extension, 61, 94, 95
Farey series, 49
finite field, 314
proof, 314
function (finite field)
proof, 314
function field, 313, 314
proof, 313
Weil, 313, 314
generalization, 107
Hardy and Littlewood, 51
Hardy’s theorem, 24, 27
Hilbert–Schmidt integral operator, 52
history, 39
hypothesis H, 44
implication, 120, 122, 133, 135, 137,140, 142, 143, 154, 155, 157
Lindelof hypothesis, 138intuitive statement, 6Lindelof hypothesis, 65Lindelof’s relaxation, 65Liouville’s function, 6, 46Mertens’ conjecture, 69Mertens’ function, 47Montgomery’s pair correlation
conjecture, 41necessary condition, 24, 27Nyman–Beurling equivalent form, 53order
symmetric group element, 54original statement, 183, 304Polya’s conjecture, 399–401partial sum, 71, 317, 320, 321, 324,
325prime number theorem, 43probabilistic argument, 121proof, 72
false, 69, 72ideas, 93
quasi, 128random matrix, 40, 42random matrix theory, 40Redheffer matrix, 53Salem, 52Skewes number
bound, 43statement affecting, 140statement implying, 57, 70, 71, 133,
138, 149, 154, 240, 244, 317, 320,321, 324, 325, 341, 350, 351, 400,451
sum of divisors function, 48variety (finite field), 103, 111
proof, 103variety (general)
proof, 125verification, 19, 24, 27, 29, 32, 34, 35,
37–39, 42, 99, 100, 112, 120, 121,125, 130, 133, 134, 241, 244, 296,351, 399, 407
zeta functionnumber field, 58, 124
zeta functions, 101proof, 101
526 Index
Riemann summability, 451, 452, 456Riemann surface, 99, 183Riemann zeta function, 4, 5, 9–12, 14,
15, 29, 31, 33, 35, 37, 94, 95,107, 116, 117, 123, 131, 161, 183,190, 199, 200, 222–224, 228, 296,297, 302, 317–319, 375, 376, 399,400, 406, 424, 425, 449, see zetafunctions, ζ(s)
L-functions, 98behavior, 69integral formula, 428partial sum, 71, 317, 320, see partial
sumprime numbers, 46properties, 183, 424, 428
Riemann’s explicit formula, 102, 119Riemann’s formula, 97, 297Riemann’s memoir, 15, 97, 132,
183–189, 303English translation, 183, 190
Riemann–Lebesgue lemma, 431Riemann–Roch theorem, 101, 103Riemann–Siegel formula, 32–34, 38, 39,
99, 120, 133, 144, 150, 153Euler–MacLaurin summation
formula, 32Riemann–Stieltjes assertation, 211Riemann–von Mangoldt formula, 132,
136Riesz, 298Riesz mean, 324, 326, see meanRiesz’s theorem, 298ringZn, 473A, 58, 108R∗, 54Z/pZ, 64OK , 58Z, 473adeles, 58, 108integers, 58, 473integers modulo n, 473polynomial, 471
RMT, see random matrix theoryRobin, 48Rogosinski, 456Rolle’s theorem, 120root, see zero
primitive, 64Rosser, 34Rosser’s rule, 34
false, 34Lehman, 34
Rouche, 348Rouche’s theorem, 348Rubinstein, 126Rudnick, 103, 128Rumely, 471running time, 469–473Ruzsa, 446
seriesDirichlet, 204, 206, 219, 298Lν(s), 208sommablilite, 298
Meyer, 220Weber, 220
Sabbagh, 71Saffari, 448, 465Sahai, 477Saias, 121Salem, 51, 52
Riemann hypothesis, 52Sarnak, 60, 98, 103, 106, 107, 125, 126,
128Saxena, 63, 110, 469, 470Schonhage, 35, 125Schaar, 231Schlomilch, 204Schlomilch function, 204Schmidt, 52, 101Schneider, 120Schwarz, 155, 414, 443, 446, 466Selberg, 27, 61, 100, 120, 122, 124,
125, 142, 144, 350, 353, 354, 364,365, 369, 371, 372, 407, 420, 425,433–435, 438–440, 447, 448, 457,459, 461
Dirichlet’s theorem, 365Selberg zeta function, 137, see zeta
functions, Z(s)Selberg’s asymptotic formula, 353, 354,
358, 363, 364, 433, 434, 438–441proof, 358, 365
Selberg’s result, 136extension, 447
Selberg’s trace formula, 122, 123
Index 527
Selberg’s upper bound sieve, 440semi-simple group, 112series∑
nµ(n)
n, 284, 442
convergence, 284, 419, 422∑p
1ps , 201, 211, 302, 304
convergence, 302∑p
log(p)p−1
, 277∑p
log(p)p
, 365, 372, 419, 422, 435, 441∑p log(p), 96, 275, 303, 354, 365∑nk=1
1ks , 317∑
pmlog(p)
pm , 268, 276, 277∑pm log(p), 268, 275, 447
Farey, 48Fn, 48
nontrivial zeros, 23, 50, 119, 122, 124,135, 231, 233–237, 252–256, 260,264, 291, 399, 400, 405–407, 409,416, 449∑, 252, 264, 291∑′, 252, 264, 291∑′′, 252, 264, 292∑ yσ−1
σ2+t2, 264∑
ρ1
(u−ρ)(1−ρ), 235–237∑
ρ1
1−ρ, 230∑
ρ1
γ2 , 378, 379∑ρ
1γ
, 378, 379∑ρ
1ρ(1−ρ)
, 231∑ρ
1ρ, 230∑
ρ1
σ2+t2, 233, 234∑
ρ1
s−ρ, 229, 240–243, 245, 246,
248, 249, 252–256, 291, 292∑ρ
1tr , 261∑
ρσ
σ2+t2, 231∑
ρyρ−1
ρ(ρ−1), 276∑
ρyσ−1
σ2+t2, 265–267, 271, 273, 275,
283∑ρ
yσ−1
t2, 260, 261, 263∑s
r, 261upper bound, 378
series representation, see Dirichletseries, see partial sum
F (X,T ), 449F (y, λ), 394L(x), 400LE(s), 124
L∆(s), 124M(x), 69, 402, 433–435, 438, 439R(t, x), 151S(t, x), 152Π(x), 377, 381<(log(ζ(s))), 16ξ′(s)ξ(s)
, 135ζ′(s)ζ(s)
, 23, 214, 216, 245, 286, 304ζ′(s)ζ2(s)
, 285
log((s− 1)ζ(s)), 96log(ξ(s)), 193log(ζ(s)), 16, 193, 200, 203
convergence, 200log(n), 433log(s− 1), 211R(log(ζ(s))), 201ψ(x), 307, 379, 427, 429, 447ψ1(x), 380θ(x), 303ϑ(x), 354, 365ξ(t), 193ζ(s), 33, 428ζ(s, a), 136Li(x), 96
Serre, 107, 110setN , 473introspective integers, 476positive integers, 473
Severi, 103, 314, 315Sherman, 352Shilov, 143Shimura, 107Shimura–Taniyama–Weil conjecture,
107Siegel, 32, 70, 120, 127, 133, 150sieve, 64, 127, 438–440, 471, see large
sieve, see upper bound sieveestimate, 450large, 438, 439prime distribution, 439prime number density, 471technique, 433
sieve of Eratosthenes algorithm, 470running time, 470
signA∗T (u), 402, 404L(x), 400
528 Index
verification, 400
λn, 50
π(x) − Li(x), 43, 134, 198, 375, 376,381, 386, 389, 397
change, 376
ξ(t), 38
change, 38
ξ(s), 27
ξ(t), 99
change, 99, 100
ζ( 12
+ it), 33
change, 33
change, 334, 335
Gauss sum, 108
simple pole
L1, 208
Z(t, C), 101
Z2(ξ), 139
Γ (s), 14
ξk, 205
ζ(σ), 17
ζ(s), 11, 14, 15, 95, 99, 117, 131, 200,304, 319
ζ(s, C), 101
simple zero
Z(t), 133
ζ(s), 14, 38, 39, 100, 133, 134, 407
Skewes, 43, 44, 375, 376
Skewes number, 43
bound, 43, 44
upper bound, 43, 44
Snaith, 67, 111, 125, 126
Sobolev space, 112
Solovay, 63, 471
Solovay’s primality testing algorithm,471
Solovay-Strassen algorithm, seealgorithms
Sophie Germain primes, 471, 479, seeconjectures
asymptotic behavior, 479
density, 471, 479
primality test, 471, 479
Soundararajan, 67
spacing between zeros
L-function, 103, 129
Z(t), 134, 135, 142, 149
ξ(t), 103
ζ(s), 41, 42, 100, 119, 124, 127, 128,133, 142, 405, 406, 449, 459
kth, 41average, 408distribution, 410normalized, 41small, 43
ζK(s), 408bound, 408
zeta function, 103spectral interpretation
zerosL-function, 112Λ(s, π), 111ζ(s), 111, 121, 122, 126
spectral theory, 139, 141non-Euclidean Laplacian, 139
spectrum, 122Spira, 136Steiner, 183Stern, 183Stieltjes, 69, 119, 200, 211, 212, 285Stirling, 21Stirling’s formula, 21, 22, 26, 135, 150Strassen, 63, 471Strassen’s primality testing algorithm,
471subconvex estimate, 111Sudan, 477sum of divisors function, 47, 48
Riemann hypothesis, 48summability, 452
Dirichlet series, 298support, 54supremum
M(n)n−12 , 70
lower bound, 70M(x)
x, 434
σn, 227, 244, 247zeros
imaginary parts, 241, 244, 408, 410surface, 103
arithmetic hyperbolic, 110projective nonsingular, 103
Swinnerton-Dyer, 128Sylvester, 225symmetric functional equation, 117, 150ζ(s), 117, 150
symmetric group, 53
Index 529
element, 53order, 53
symmetric matrix, 405, 409eigenvalue, 405, 409
symplectic matrix, 103, 126, 405, 409eigenvalue, 103, 405, 409unitary, 126
Szele Prize, 317
Taniyama, 107Tate, 104Tauberian character, 451Tauberian theorem, 308, see auxiliary
Tauberian theoremTauberian theory, 426Taylor, 99Taylor series, 16, 26, 305, 328Taylor’s formula, 144, 151Tchebotarev’s density theorem, 98Tchebychev, 97, 212, 225, 226, 420, 422,
see Chebyshevte Riele, 44, 62, 69, 70, 100, 119, 133,
134, 139ternary additive divisor problem, 141Terras, 123theoreme
Bohr, 298Cauchy, 297Halphen, 212, 218, 219nombres premiers, 433, 434
demonstration, 434Riesz, 298von Mangoldt, 260, 261
theoremsI1(T ) ∼ log T , 67
proof, 67I2(T ) ∼ 1
2π2 (log T )4, 67proof, 67
Ik(T ) ≥ (a(k) + o(1))(log T )k2, 67
conditional proof, 67proof, 67
Ik(T ) ≥ 2(ak + o(1))(log T )k2, 67
proof, 67M(x) = o(x), 307, 308, 433, 434, 438,
439proof, 311, 433, 434, 438–440, 442
N(T ) ∼ T2π
log( T2π
) − T2π
, 15, 19, 24,31, 41
proof, 15, 24, 31
von Mangoldt, 15, 311
Γ (µ)
∑p<x log(p) logµ−1( x
p) ∼ x, 218
π(x) ∼ xlog(x)
, 16, 162, 172, 302, 309,420, 425, 427, see prime numbertheorem
π(x) ∼ Li(x), 43, 46, 61, 96, 98, 132,134, 162, 199, 222, 225–227, 282,284
proof, 199, 222ψ(x) ∼ x, 305, 308, 427
proof, 312, 429∑n<x Λ(n) ∼ x, 119, 307∑n<x
µ(n)n
= O(1), 442∑n<x
µ(n)n
= o(1), 288, 307, 308proof, 312∑p<x
log(p)p
∼ log(x), 278, 365, 419,435, 441
proof, 422, 423∑p<x log(p) ∼ x, 212, 218, 275∑p≤x
log(p)p
= log(x) +O(1), 354∑pm<x
log(p)pm ∼ log(x), 277∑
pm<x log(p) ∼ x, 275θ(x) ∼ x, 303ϑ(x) ∼ x, 354, 356, 365
Time magazine, 70Titchmarsh, 58, 63, 65, 66, 100, 121,
127, 296, 307, 388, 426, 459Titchmarsh’s S(T ) function, 65trace
Frobenius endomorphism, 102trace formula
Selberg, 122, 123
transcendence of 2√
2, 120trivial zero, 14, 95, 118ζ(s), 14, 131, 191, 228, 304, 319
Tsang, 142, 148Turan, 71, 134, 317, 318, 402, 440, 445,
446Turan’s conjecture, 402, 483
conditional disproof, 402Turan’s inequality, 402, see Turan’s
conjectureconditional disproof, 402disproof, 404
Turan–Kubilius inequality, 440, 445,446
dual, 445, 446large sieve inequality, 440, 445, 446
530 Index
Turing, 34, 473Turing machine, 473
deterministic polynomial time, 473Turing’s algorithm, 34twin prime constant, 479twist, 126
quadratic, 126
Ungar, 320Ungar’s theorem, 320unitary linear operator, 405, 409unitary matrix, 67, 103, 123, 405, 409
characteristic polynomial, 67eigenvalue, 103, 123, 405, 409symplectic, 126
unitary representationirreducible, 108
upper boundE1(T ), 138E2(T ), 138Ek(T ), 138, 140Ik(T ), 66M(x), 440S(T ), 142S(t), 112SM (T ), 156Z(t), 143∆k(T ), 140π(x), 96, 97, 303, 429π2(x, k), 458ψ(x), 97, 379, 429σ, 227, 244, 247, 375–377, 380σ(n), 48σ(y), 267, 268∑
ρ1
γ2 , 378∑ρ
1γ
, 378∑ρ
yσ−1
σ2+t2, 265–267∑
ρyσ−1
t2, 263, 264
ϕ(x), 168ϑ(x), 97, 355, 366ζ( 1
2+ it), 138
Li(x), 380, 381cardinality of G, 477distribution of zeros, 412infimumM(n)n−
12 , 70
prime distribution, 439, 440Skewes number, 43, 44
zerosDavenport–Heilbronn zeta function,
136imaginary parts, 416
upper bound sieve, 440Selberg, 440
Vaaler, 457valeur principale
de Li(x), 226valueN1(T ), 38S(T ), 142ζ′(s), 49ζ( 1
2+ it), 142
van de Lune, 100, 120, 134van der Waerden, 314variety
algebraic, 100, 101arithmetic, 98field, 103finite field, 103, 111
Riemann hypothesis, 103, 111general, 125
Riemann hypothesis, 125non-singular, 103projective, 103zeta functions, 103
Vaughan, 448, 465Verdier, 101Vinogradov, 18, 62, 110, 322Vinogradov’s theorem, 62Volchkov, 52von Mangoldt, 15, 31, 132, 210, 225,
226, 229, 260, 261, 285, 434von Mangoldt’s function, 119, 428, 434,
see arithmetic functions, Λ(n)von Mangoldt’s theorem, 260, 261
waveformMaass, 99, 104primitive, 99
Weber, 220Weber series, 220Weber’s formula, 220Wedeniwski, 120Weierstrass, 422Weierstrass M -test, 422Weil, 101–104, 107, 125, 126, 313, 314
Index 531
Weil conjecture, 106, 111, 313Weil’s explicit formula, 102, 122, 123L-function
number field, 102ζ(s), 102zeta function
curve, 102Weil’s positivity criterion, 122Weil–Guinand–Riemann formula, 112Weinberger, 408Weyl, 120, 322, 336Weyl sum, 120Weyl’s inequality, 336Wiener, 420, 424, 426, 452Wiener theory, 426
prime number theorem, 426Wiener’s Tauberian theorem, 452Wiener’s Tauberian theory, 426Wiles, 4, 99, 120, 124, 129Wilkins, 183, 190Wilson, 301Wilson’s theorem, 301Winter, 100Winther, 351Wirsing, 439Wirsing’s mean value theorem, 439
Xi function, 51, 297xi function, 14, 19–22, 25–27, 31, 38, 50,
192, 297Xian-Jin Li, 122
Yau, 313
zeroLν(s), 209ζ(s), 199–201, 214, 226, 228, 229, 296,
297conjugees, 229, 230distribution, 199, 200imaginaires, 227, 228, 233, 240
Zagier, 128zero
(1− 21−2s)ζ(2s), 298location, 298
H(λ, z), 51real, 51
Jn(s), 329location, 329
L-function, 98, 100, 103, 112, 126,127, 129, see Landau–Siegel zero
critical line, 60, 126, 129critical strip, 126distribution, 111, 112, 126elliptic curve, 129fluctuation, 111high multiplicity, 128, 129location, 98multiplicity, 128triple, 128
L(s, χ)line <(s) = 1, 350location, 350
L(s, χ3), 123, 124L(s, χk), 126, 127
connection, 127L(s, π)
line <(s) = 1, 110LE(s), 124Lν(s), 209, 210
conjugate, 209distribution, 209, 210location, 209
MZ,f (t), 144, 147, 154, 156γ, 147, 156distribution, 144, 147, 154location, 144multiplicity, 144
Rn(s)asymptotic behavior, 349
Un(s), 321, 322, 326condensation point, 322, 326location, 322, 335
Vn(s), 325, 326, 347, 349asymptotic behavior, 325, 347condensation point, 325, 326, 347location, 343, 349
Z(t), 133–135, 142–144, 146, 149, 154γn, 135, 142distribution, 143, 144, 146location, 144multiplicity, 133, 143, 144simple, 133
Z(t, C), 101Λ(s), 107
location, 107Λ(s, χ), 112
critical line, 112
532 Index
eigenvalue, 112location, 112
Λ(s, π), 109, 111fluctuation, 111location, 109, 111
Ξ(t), 121η(s), 49, 56
critical line, 49critical strip, 491
ζ(s), 285, 288
sin θ(t), 33ξ′(s), 120, 121ξ( 1
2+ it), 121
ξ(t), 38multiplicity, 38odd, 38
ξ(s), 15, 19, 27, 31, 38, 95argument principle, 20critical strip, 19distribution, 95imaginary part, 38location, 38
ξ(t), 103, 193distribution, 193location, 193multiplicity, 99multiplicity odd, 99
ζ( 12
+ it), 119, 212ζ(s), 14, 15, 24, 31, 95, 100, 109, 111,
119–125, 127, 128, 133, 157, 191,199–201, 212, 214, 224, 226, 228,229, 240, 244, 296, 297, 304, 319,376, 405, 406
ρ, 95, 376, 378ρn, 399, 400k-tuple correlation, 409behavior, 113, 133, 211, 405, 407,
409computation, 32conjugate, 119, 209, 227, 230, 247,
320connection, 127counting, 18critical line, 5, 15, 18, 24, 27, 30–32,
34, 35, 38, 40, 100, 112, 119, 120,125, 132–134, 157, 212, 240, 244,296, 297, 307, 400, 407
critical strip, 18, 19, 24, 30, 41, 297,304, 378, 426
distribution, 41, 65, 95, 99, 118,119, 125, 127, 136, 138, 157, 199,200, 260, 303, 319, 378, 407, 412,459
eigenvalue, 121, 123, 125imaginary part, 40, 41, 133, 376,
378, 387, 388, 399–401, 405–407,447, 449, 456
integer part, 124line <(s) = 1, 16–18, 61, 110, 119,
199, 201–203, 221, 222, 224, 226,304, 305, 420, 425, 428
linear dependence, 400linear operator, 405, 409location, 14, 15, 24, 30, 70, 95, 99,
100, 112, 119, 120, 131, 132, 191,199, 200, 228, 240, 244, 260, 296,297, 304, 319, 320, 375–378, 380,399, 401, 403, 404, 412
multiplicity, 39, 100, 122, 128, 133,134, 400, 407–409, 416
pair correlation, 447, 448, 459prime number, 410relation between σ and t, 120, 240,
244, 250, 251, 257–259, 289, 294,295
simple, 14, 38, 39, 100, 133, 134, 407symmetric, 15, 119
ζK(s), 58, 408critical line, 408critical strip, 408
alternating zeta function, 325wk, 325, 326, 343, 347, 349line <(s) = 1, 325, 350
Artin L-function, 100Bessel function, 129convolution function, 134Davenport–Heilbronn zeta function,
136asymptotic behavior, 136behavior, 136critical line, 136critical strip, 136distribution, 136location, 136
Dirichlet L-function, 100Dirichlet series, 328elliptic curve L-function, 129
multiplicity, 129
Index 533
Epstein zeta function, 136critical strip, 136
family, 111L-functions, 111
Landau–Siegel, 127, 128Maass L-function, 100meromorphic function, 19
counting, 19partial sum, 71, 317, 327, 328
behavior, 322condensation point, 327, 328, 352Dirichlet series, 328location, 71, 317Riemann zeta function, 71
Riesz mean, 349behavior, 349condensation point, 349
zeta function, 111, 125critical line, 111distribution, 125eigenvalue, 111
zeta functions, 101, 103zero-density theorem, 66, 110, 111zero-free region, 16
(z − 1)ζ(s), 420Cn(s), 324Jn(s), 329L-functions, 112L(s, χ), 350L(s, π), 110Un(s), 320–323, 338Vn(s), 325π(x)− Li(x), 375ξ(t), 193ζ′(s), 50ζ(s), 15, 16, 18, 110, 118, 119, 199,
200, 203, 222, 319–321, 420, 428Korobov, 18Vinogradov, 18
Dirichlet series, 210partial sum, 71, 317
Riemann zeta function, 71zeta functions, 35, 101, 123, 124, 132,
136, see alternating zeta func-tion, see Davenport–Heilbronnzeta function, see Epstein zetafunction, see Selberg zeta function
(1 − 21−s)ζ(s), 325, see alternatingzeta function
(z − 1)ζ(s), 420
Z(t, C), 101
degree, 101
numerator, 101
rational function, 101
=(ζ(s)), 118
<(ζ(s)), 118
η(s), 35, 36, 56, see alternating zetafunction, see Dirichlet eta function
ζ(σ), 17
ζ(s), 4, 5, 10–12, 14, 15, 29, 31,33, 35, 56, 65, 66, 95, 99, 107,117, 131, 132, 138, 190, 200, 201,222–224, 228, 296, 297, 302, 319,376, 400, 425, see Riemann zetafunction
L(s, χ1), 56
ζK(s), 58
s = 0, 280
behavior, 34, 66
critical line, 34
derivative, 49, 99
integral formula, 97
logarithm, 16, 97, 193, 200, 201, 304
logarithmic derivative, 22, 214, 216,229, 304, 427, 428
partial sum, 320
peak, 40
properties, 304
ζ(s, C), 101
ζ(s, a), 136, see Hurwitz zeta function
ζk(s), 140
ζK(s), 58, 315, 316, 408, see Dedekindzeta function
numerator, 315
f(s), see Davenport–Heilbronn zetafunction
Z(s), 137, see Selberg zeta function
(1− 21−2s)ζ(2s), 298
cohomology, 103
curve, 101, see zeta functions, ζ(s, C)
Dirichlet, 107, see L-functions
field, 103
finite field, 125, 126
history, 95
number field, 124
variety, 103
Zinoviev, 62