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Авторизация |
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Apostol T.M. — Introduction to Analytic Number Theory |
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Предметный указатель |
0, big oh notation 53
Abel, Niels Henrik 77
Abelian group 129
Abel’s identity 77
Abscissa of absolute convergence 225
Abscissa of convergence 233
Additive number theory 304
Algorithm, division 19
Algorithm, division, Euclidean 20
Analytic continuation, of Dirichlet L-functions 255
Analytic continuation, of Dirichlet L-functions of Hurwitz zeta function 254
Analytic continuation, of Dirichlet L-functions of Riemann zeta function 255
Apostol, Tom M. 329
Arithmetic, fundamental theorem of 17
Arithmetical function 24
Asymptotic equality 53
Average (arithmetic mean) of an arithmetical function 52
Average order 52
Ayoub, Raymond G. 329
Bell series 43
Bell, Eric Temple 29 42 329
Bernoulli, numbers 265
Bernoulli, numbers, periodic functions 267
Bernoulli, numbers, polynomials 264
Binomial congruence 214
Borozdkin, K. G. 10 329
Buhstab, A. A. 11 329
Cauchy product 44
Cauchy, Augustin-Louis 44 144 198
Chandrasekharan, Komaravolu 329
Character, Dirichlet 138
Character, Dirichlet, of an abelian group 133
Character, Dirichlet, primitive 168
Character, Dirichlet, principal 138
Chebyshev function 75
Chebyshev function 75
Chebyshev, Pafnuti Liwowich 9 75
Chen, Jing-run 11 304 329
Chinese remainder theorem 117
Clarkson, James A. 18 330
Classes of residues 109
Clausen, Thomas 275
Common divisor 14
Commutative group 129
Complete residue system 110
Completely multiplicative function 33
Conductor of a character 171
Congruence 106
Convolution, Dirichlet 29
Convolution, Dirichlet, generalized 39
Critical line 293
Critical strip 293
Cross-classification principle 123
Cyclic group 131
Darling, H. B. C 324
Davenport, Harold 330
Decomposition property of reduced residue systems 125
Derivative of arithmetical functions 45
Dickson, Leonard Eugene 12 330
Diophantine equation 5 190
Dirichlet, character 138
Dirichlet, character , convolution (product) 29
Dirichlet, character , divisor problem 59
Dirichlet, character , estimate for d(n) 53 57
Dirichlet, character , inverse 30
Dirichlet, character , L-function 224
Dirichlet, character , series 224
Dirichlet, character , theorem on primes in arithmetic progressions 7 146 154
Dirichlet, Peter Gustav Lejeune 5 7 29 53 138 146 224
Disquisitiones Arithmeticae 5
Divisibility 14
Division algorithm 19
Divisor 14
Divisor function 38
Edwards, H. M. 330
Elementary proof of prime number theorem 9 98
Ellison, W. J. 306 330
Erdos, Paul 9 330
Euclidean algorithm 20
Euclid’s lemma 16
Euler product 230
Euler totient function 25
Euler — Fermat theorem 113
Euler's summation formula 54
Euler, Leonhard 4 5 7 9 19 25 53 54 113 180 185 230 308 312 315
Euler’s constant 53 250
Euler’s criterion 180
Euler’s pentagonal-number theorem 312
Evaluation of 181
Evaluation of 181
Evaluation of 268
Evaluation of 264
Evaluation of 266
Exponent of a modulo m 204
Exponential congruence 215
Factor 14
Fermat 7
Fermat conjecture 11
Fermat prime 7
Fermat theorem (little) 114
Fermat, infinitude of 16 19
Fermat, Mersenne 4
Fermat, Pierre de 5 7 11 113 114
Finite Fourier expansion 160
Formal power series 41
Fourier coefficient 160
Franklin, Fabian 313 330
Function, 8 74
Function, 278
Function, arithmetical 24
Function, Bernoulli periodic 267
Function, Chebyshev 75
Function, Chebyshev 75
Function, completely multiplicative 33
Function, Dirichlet 224
Function, divisor , 38
Function, Euler totient 25
Function, Hurwitz zeta 249
Function, k(n) 247
Function, Liouville 37
Function, M(x) 91
Function, Mangoldt 32
Function, Mobius 24
Function, periodic zeta F(x, s) 257
Function, Riemann zeta 9 249
Function, v(n) 247
Functional equation for 263
Functional equation for 259
Functional equation for 261
Functional equation, for 250
Fundamental Theorem of Arithmetic 17
Gamma function 250
Gauss sum, associated with 165
Gauss sum, associated with , quadratic 177 306
Gauss, Carl Friedrich 5 7 8 106 165 177 182 185 306 326
Gauss’ Lemma 182
Gauss’ triangular-number theorem 326
Generating function 308
Geometric sum 157 158
Gerstenhaber, Murray 186 330
Goldbach conjecture 9 304
Goldbach, C 6 9 304
Greatest common divisor 15 20 21
Greatest integer symbol 8 25 54 72
Grosswald, Emil 330
Group character 133
Group, abelian 129
Group, cyclic 131
Group, definition of 129
Hadamard, Jacques 9 74 330
| Hagis, Peter, Jr. 5 330
Half-plane, of absolute convergence 225
Half-plane, of convergence 233
Hardy, Godfrey Harold 59 293 305 330
Hemer, Ove 330
Hilbert, David 293
Hurwitz formula for 257
Hurwitz zeta function 249 251 253 255
Hurwitz, Adolf 249
Identity element 30 129
Identity function I(n) 30
INDEX 213
Index calculus 214
Indices (table of) 216 217
Induced modulus 167
Induction, Principle of 13
Inequalities for 1 272
Inequalities for 270 287 291
Inequalities for 270
Inequalities for 82
Inequalities for 298
Inequalities for d(n) 294
Inequalities for nth prime 84
Inequalities for p(n) 316 318
Infinitude of primes 16 19
Ingham, A. E. 330
Inverse, Dirichlet 30
Inverse, Dirichlet, of completely multiplicative function 36
Inversion formula, Mobius 32
Inversion formula, Mobius, generalized 40
Iseki, Kaneshiro 99 332
Jacobi symbol 188
Jacobi triple product identity 319
Jacobi, Carl Gustav Jacob 187 305 313 319
Jordan totient 48
Kloosterman sum 176
Kloosterman, H. D. 176
Kolberg, Oddmund 324
Kolesnik, G. A. 59
Kruyswijk, D. 324 327 331
L-function 224
Lagrange interpolation formula 158
Lagrange, Joseph Louis 5 115 144 158
Lagrange’s theorem on polynomial congruences 115
Landau, Edmund 59 237 248 301 331
Landau’s theorem 237 248
Lattice points 57 62
Lattice points, visibility of 62
Law of quadratic reciprocity 185 189 193 200
Least common multiple 22
Leech, John 10 331
Legendre symbol 179
Legendre, Adrien-Marie 5 67 179 185 331
Legendre’s identity 67
Lehmer, Derrick Henry 6 293 316 331
Lehmer, Derrick Norman 6 331
Lemma of Gauss 182
LeVeque, William Judson 2 191 331
Levinson, Norman 293 331
Linear congruence 111 112 114 214
Liouville function 37
Liouville, Joseph 37
Little Fermat theorem 114
Littlewood, John Edensor 10 305 331
Logarithmic integral Li(x) 102
Lucas, Edouard 275
MacMahon, Percy A. 316
Mean value formulas for Dirichlet series 240
Mersenne numbers 4
Mersenne, P. 4
Mertens, Franz 91
Mertens’ conjecture 91
Mills, W. H. 8 331
Mobius function 24
Mobius function of order k 50
Mobius inversion formula 32
Mobius inversion formula, generalized 40
Mobius inversion formula, product form 47
Mobius, Augustus Ferdinand 24
Mordell, Louis Joel 305 306
Multiplication, Dirichlet 29
Multiplication, Dirichlet of residue classes 138
Multiplicative function 33
Multiplicative number theory 304
Nevanlinna, Veikko 331
Niven, Ivan 331
Nonresidue 178
Number-theoretic function 24
o, little oh notation 94
Order of a group 130
Orthogonality relation for Dirichlet characters 140
Orthogonality relation, for group characters 137
Partition 304
Partition function p(n) 307
Pentagonal numbers 2 5 311
Pentagonal-number theorem 312
Perfect numbers 4
Periodic arithmetical function 157
Periodic zeta function 257
Polya inequality for character sums 173 176 299
Polygonal numbers 2 5
Polynomial congruence 115
Prachar, Karl 331
Prime number theorem 9 65 74 79 92 94 98 278 289
PRIMES 2 16
Primes in arithmetic progressions 7 146 154
Primes, contained in a factorial 67
Primitive character 168
Primitive root 204
Principal character 134 138
Product of Dirichlet series 228
Product, of arithmetical functions 29
Pythagorean triple 2
Quadratic, congruence 178
Quadratic, Gauss sum 177 195
Quadratic, nonresidue 178
Quadratic, reciprocity law 185 189 193 200
Quadratic, residue 178
Rademacher, Hans 316 331
Ramanujan partition identities 324 328
Ramanujan sum 160 176
Ramanujan, Srinivasa 160 305 324 328
Reciprocity law for Legendre symbols 185 193 200
Reciprocity law for quadratic Gauss sums 200
Reciprocity law, for Jacobi symbols 189
Reduced fraction 21
Relatively prime 15 21
Relatively prime in pairs 21
Renyi, Alfred 10 332
Residue class 109
Residue system, complete 110
Residue system, reduced 113 125
Residue, quadratic 178
Riemann hypothesis 293 301
Riemann zeta function 249
Riemann — Lebesgue lemma 279
Riemann — Stieltjes integral 77
Riemann, Georg Friedrich Bernhard 9 225 293 332
Ring of formal power series 42
Robinson, Raphael M. 7 332
Rosser, J. Barkley 293 332
Schnirelmann, L. 10 332
Selberg asymptotic formula 100 103 104
Selberg identity 46
Selberg, Atle 9 46 100 332
Separable Gauss sums 165 171
Shanks, Daniel 312 326 332
Shapiro, Harold N. 85 146 332
Shapiro’s Tauberian theorem 85
Shen, Mok-Kong 9 332
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