Edwards H.M. — Riemann's Zeta Function :: Электронная библиотека попечительского совета мехмата МГУ

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Edwards H.M. — Riemann's Zeta Function

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Название: Riemann's Zeta Function

Автор: Edwards H.M.

Аннотация:

Edwards elaborates on Bernard Riemann's eight-page paper On the Number of Primes Less Than a Given Magnitude, published in German in 1859. His goal is not to supplant the classic work, but to provide mathematics students access to it. Indeed an English translation of the original is appended. Academic Press published the 1974 edition.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 330

Добавлена в каталог: 04.05.2008

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Предметный указатель

Abel integral formula for $\zeta(s)$       9 221
Abel's theorem      278
Alpha notation for roots $\rho=1/2+ia$       36
Analytic continuation of a particular integral      276—277
Analytic continuation of zeta function      11 16n 115
Analytic continuation, Riemann's conception of      9
Approximate functional equation      Hardy and Littlewood
as consequence of Riemann hypothesis      188
Asymptotic expansions      Euler-Maclaurin summation for evaluation of zeta Logarithmic Riemann Stirling's 87
Backlund      97
Backlund determination of N(T) for T=200      128—129
Backlund estimation of N(T)      132—134
Backlund theorem on Lindelof hypothesis      182 188—190
Bernoulli numbers      11 103
Bernoulli numbers $B_{2n+1}=0$       11n 14n 103
Bernoulli polynomials       100—103
Bernoulli polynomials periodified $\overline{B}_n(x)$       104
Bohr and Landau Riemann hypothesis implies S unbounded      181 201—202
Bohr and Landau theorem on roots near Re s=1/2      19 193—195
Bombieri      297
Cesaro averages      279
Chebyshev estimates of $\pi$ from estimates of $\psi$       68 76—77
Chebyshev estimation of $\pi (x)$       3—4 281—284
Chebyshev mentioned by Riemann      5
Chebyshev's identity      281—284
Conditional convergence in Fourier analysis      23
Conditional convergence of sums over $\rho$       20—21 30 35 49—50
DE LA VALLEY POUSSIN estimate of error in prime number theorem      78—95
DE LA VALLEY POUSSIN proof of prime number theorem      68
DE LA VALLEY POUSSIN proof that $\sum \mu(n)/n=0$       91—95
Debye      139
Definite integrals      Riemann evaluation
Denjoy      268—269
Dirichlet      298 299
Dirichlet acquaintanceship with Chebyshev      4
Dirichlet use of Euler product formula      7
Erdos      282 288
Euclid      1
Euler $\phi$ -function      250—251
Euler $\Sigma p^{-1}$ diverges      1
Euler and factorial function      7—8
Euler and functional equation of zeta      12
Euler formula $\sum^{\infty}_{n=-\infty} x^{n}=0$       212
Euler product formula      Euler 1 6—7 22—23 50n
Euler statement of $\Sigma \mu(n)/n=0$       92
Euler — Maclaurin summation      Stirling's series 97
Euler — Maclaurin summation for evaluation of zeta      114—115
Euler — Maclaurin summation statement of method      98—106
Euler — Maclaurin summation summary of method      106
Euler's constant      67 106n
Explicit formulas      J function psi
Factorial function      Stirling's series 7—9
Farey series      263—264
Farey series related to Riemann hypothesis      264—267
Fourier analysis      23—25 203—225
Fourier analysis adjoint of an operator      205
Fourier analysis inversion formulas      24 27 51 54—56 205 213—215
Fourier analysis transform of an (invariant) operator      204
Fourier analysis, interpretation of Chebyshev's identity      282—283
Fourier series of $\overline{B_1}(x)$       196
Fourier transform      Fourier analysis transform 209 211
Franel and Landau Farey series and the Riemann hypothesis      263—267
G function      206
G function functional equation      209—210
Gamma function      Factorial function
Gauss      299 305
Gauss counts of primes      2 305
Gauss notation $\prod$ for factorial      8
Gauss on density of primes      2
Gram computation of 15 roots $\rho$       96—97
Gram points      125—126
Gram's law      127 171
Gram's law exceptions      126—127 176n
Gram's law statement of      126
H function      207
Hadamard proof of prime number theorem      6 38 68
Hadamard proof of product formula for $\xi$       18 21 39
Hadamard proof that $\zeta(l+it)\neq 0$       69—72
Hadamard publication of three circles theorem      187
Hardy      136
Hardy and Littlewood approximate functional equation      201n 229n
Hardy and Littlewood average of $(|\zeta(s)|)^{3}$ on Re=const      195
Hardy and Littlewood estimate of $\zeta(1/2)$       201
Hardy and Littlewood KT roots on Re s=1/2      19 226 229—237
Hardy and Littlewood reformulation of Lindelof hypothesis      201
Hardy and Littlewood Tauberian theorems for Cesaro averages      279
Hardy and Littlewood use of Tauberian theorem to prove prime number theorem      280
Hardy infinitely many roots on Re s=1/2      19 226—229
Haselgrove      96 121 157 161 178
Haselgrove excerpts from tables      122—123 158
Haselgrove on exceptions to Gram's law      126n
hilbert      6 298
Hutchinson      97
Hutchinson numerical analysis of roots $\rho$       126—127 129—132
Ikehara's theorem      281
J function      22
J function in terms of $\pi(x)$       33
J function Riemann's formula for      33 48 61—65
jacobi      15
Jensen's theorem      39—41
KUZMIN proof of Riemann — Siegel integral formula      273—278
Landau      62 62n 136
Landau average of $(|\zeta(s)|)^2$ on Re=const      195
Landau o, O notation      Bohr and Landau Franel 200
Legendre notation for gamma function      8
Legendre on density of primes      3
Legendre relation for factorial function      9
Lehman      269
Lehman verification of Riemann hypothesis to g_{250000}      172
Lehmer's phenomenon      179
Lehmer, D. H. computations of roots $\rho$       175—179
Lehmer, D. H. on Riemann's formula for $\pi(x)$       35
Lehmer, D. H. verification of Riemann hypothesis to g_{250000}      172
Lehmer, D. N. counts of primes      3
Levinson      288
Li(x)      Logarithmic integral
Lindelof estimates of growth of $\zeta(s)$       182—186
Lindelof hypothesis      186 177n 188 201
Lindelof's theorem      184
Lindelof's theorem modified      186
LITTLEWOOD $\int^T_0 S(t)dt=O(log T)$       173
LITTLEWOOD $\pi(x)<Li(x)$ fails      269
LITTLEWOOD improvement of $\beta<1$       200
LITTLEWOOD improvement of Bohr — Landau theorem      195
LITTLEWOOD improvement of Tauber's theorem      279
LITTLEWOOD Riemann hypothesis and growth of M      261
LITTLEWOOD use of three circles theorem      Hardy and Littlewood 187
Logarithmic integral Li(x)      26
Logarithmic integral Li(x) asymptotic formula for      86
Logarithmic integral Li(x) estimate of as $x\longrightarrow\infty$       90
Logarithmic integral Li(x) value at $x^{\beta}$ for complex $\beta$       30
M function (sum of Mobius $\mu$ )      260
Maclaurin      Euler-Maclaurin summation

mellin      25n
Mellin estimate of $\zeta(l+it)$       183
Mertens proof that $\zeta(l+it)\neq 0$       79—80
Mertens's theorem      6
Mobius inversion      34 217—218 283 285
Mu function of Lindelof $\mu(\sigma)$       186
Mu function of Mobius $\mu(n)$       34 91—92 217
N function      128
N function Backhand's verification of Riemann's estimate      132—134
N function evaluated by Turing's method      172—175
Nonsense      212 217
O, o notation      200
Parseval's equation      215—216
Pi function $\pi(x)$       4 33
Pi function $\pi(x)$ approximations to      84—91
Pi function $\pi(x)$ in terms of J (x)      Prime number theorem Poisson summation formula 34 209—210
Polya theorems on functions with zeros on Re s=1/2      269—273
Prime number theorem      4
Prime number theorem improved remainder      84 200
Prime number theorem proof      68—77
Product formula for $\xi(s)$       20—21
Product formula for $\xi(s)$ proof      46—47
Product formula for sine      9 18 47 224
Psi function $\psi(x)$ of Chebyshev      49
Psi function $\psi(x)$ of Chebyshev, von Mangoldt's formula for      Chebyshev 49—61
Ramanujan's formula      218—225
Rho      Roots $\rho$
RIEMANN "everywhere valid" formula for $\zeta(s)$       9—11
RIEMANN analytic functions treated globally      20
RIEMANN comments on $\pi(x)\sim Li(x)$       34—36 269 305
RIEMANN computations of roots      159—162
RIEMANN error involving $\xi(0)$       31
RIEMANN estimate of N(T)      18—19 301
RIEMANN evaluation of definite integrals      12—13 19 26—33 146—148
RIEMANN explicit formula for J(x) (=f(x))      33 304
Riemann hypothesis      6 19
Riemann hypothesis in light of Riemann — Siegel formula      164—166
Riemann hypothesis probabilistic interpretation      268—269
Riemann hypothesis, implies Lindelof hypothesis      188
Riemann hypothesis, related to error in prime number theorem      88—91
Riemann hypothesis, related to Farey series      263—267
Riemann hypothesis, related to growth of M(x)      260—263
Riemann hypothesis, verified to $T=g_{1040}$       171
Riemann hypothesis, verified to $T=g_{250000}$       172
Riemann hypothesis, verified to $T=g_{25000}$       172
Riemann hypothesis, verified to $T=g_{3500000}$       172 179—180
Riemann hypothesis, verified to T=200      129
Riemann hypothesis, verified to T=300      129—132
RIEMANN introduction of function $\xi$       16
RIEMANN manuscript with statement of asymptotic formula      156—157
RIEMANN paper on $\pi(x)$       1—38
RIEMANN proofs of the functional equation      12—16 166—170 274 300—301
RIEMANN questions unresolved by      37—38
RIEMANN skill as analyst      136
RIEMANN statement of Riemann hypothesis      19 30n 301
RIEMANN translation of paper on $\pi(x)$       299—305
RIEMANN use of "Fouriers theorem"      23—25 302—303
RIEMANN use of saddle point method      139n
RIEMANN view of analytic continuation      9 20
Riemann — Siegel asymptotic formula      136—164
Riemann — Siegel asymptotic formula error estimates      162—164
Riemann — Siegel asymptotic formula manuscript      156—157
Riemann — Siegel asymptotic formula statement      154
Riemann — Siegel integral formula      137 166—170
Riemann — Siegel integral formula proof by Kuzmin      273—278
Riemann's estimate      18—19 301
Riemann's estimate of density      Conditional convergence Bohr 21 43 302
Roots $\rho$ of $\xi(s)=0$       18—19
Roots $\rho$ of $\xi(s)=0$ computations of      96 157—162 178
Roots $\rho$ of $\xi(s)=0$ crude estimate of density      42—43
Roots $\rho$ of $\xi(s)=0$ real parts<1      1 70—72 79—81 200
Roots $\rho$ of $\xi(s)=0$ von Mangoldt's estimate of density      56—58
Rosser's rule      180—181
Rosser, YOHE, and SHOENFELD error estimate for Riemann — Siegel formula      163
Rosser, YOHE, and SHOENFELD use of Turing's method      175 180
Rosser, YOHE, and SHOENFELD verification of Riemann hypothesis to $T=g_{3500000}$       179—180
S function S(T)      173
S function S(T) estimates of      174 190—193 201—202
Saddle point method      139—140
Selberg elementary proof of prime number theorem      282 288—297
Selberg KT log T roots on Re s=1/2      19 226 237—259
Selberg's inequality      284—288
Self-reciprocal operators      210—211
Siegel discovery of Riemann — Siegel formula      136
Siegel discovery of Riemann — Siegel formula proof that Riemann — Siegel formula is asymptotic      163
Slit plane      111
Steepest descent, method of      139—140
STIELTJES alleged proof of Riemann hypothesis      262—263
STIELTJES estimate of remainder in Stirling's series      112
STIELTJES evaluation of the constant in Stirling's series      113
Stirling      109
Stirling's formula      Stirling's series
Stirling's series      106—114
Stirling's series for $\prod'(s)/\prod(s)$       113
Stirling's series statement      109
Stirling's series Stieltjes's estimate of the remainder      112
Tauber      278
Tauberian theorems      278—281
Theta function $\Theta$ of Chebyshev      76 288
Theta function $\upsilon(t)$ in $\zeta(1/2+it)=Z(t)$ exp i $\upsilon(t)$       119
Theta function $\upsilon(t)$ in $\zeta(1/2+it)=Z(t)$ exp i $\upsilon(t)$ asymptotic formula for      120—121
Theta function of Jacobi      15 170 227
Three circles theorem      187
TlTCHMARSH as secondary source      199 226
TlTCHMARSH average of $\zeta(1/2+ig_n)$ is 2      229
TlTCHMARSH error estimate for Riemann — Siegel formula      162—163
TlTCHMARSH verification of Riemann hypothesis to T=g_1040      171
Turing's method of evaluating N(T)      172—175
Vinogradov's estimate of $\zeta(l+it)$       200
von Mangoldt estimate of N(T)      173
von Mangoldt proof of formula for $\psi$       50—61
von Mangoldt proof of Riemann's estimate of N(T)      133
von Mangoldt proof of Riemann's formula for J(x)      48 61—65
von Mangoldt proof that $\sum \mu(n)/n$       0 92
von Mangoldt statement of formula for $\psi$       49 54 66
Weil      298
Weyl's estimates of $\zeta(l+it)$       200
Wiener's Tauberian theorem      280—281
Wirsing      288 294 297
Xi function $\xi(s)$       16—18
Xi function $\xi(s)$ as transform of self-adjoint operator      206—213
Xi function $\xi(s)$ rate of growth      41
Z function Z(t)      119
Z function Z(t) asymptotic formula for      Riemann — Siegel asymptotic formula 119
Zeta function analytic continuation of      9—11 16n
Zeta function as Fourier transform of summation operator      204
Zeta function evaluation by Euler — Maclaurin summation      114—118
Zeta function growth in strip $0\leq Re\ s\leq 1$       182—201
Zeta function growth related to zeros      182 188 190 193 200
Zeta function value of $\zeta'(0)/\zeta(0)$       66—67 134—135
Zeta function values at integers      11—12
Zeta function values at negative integers      216—217
Zeta function, functional equation      Euler Riemann 12—16 222 224—225

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