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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