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Apostol T.M. — Introduction to Analytic Number Theory
Apostol T.M. — Introduction to Analytic Number Theory



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Название: Introduction to Analytic Number Theory

Автор: Apostol T.M.

Аннотация:

This introductory textbook is designed to teach undergraduates the basic ideas and techniques of number theory, with special consideration to the principles of analytic number theory. The first five chapters treat elementary concepts such as divisibility, congruence and arithmetical functions. The topics in the next chapters include Dirichlet's theorem on primes in progressions, Gauss sums, quadratic residues, Dirichlet series, and Euler products with applications to the Riemann zeta function and Dirichlet L-functions. Also included is an introduction to partitions. Among the strong points of the book are its clarity of exposition and a collection of exercises at the end of each chapter. The first ten chapters, with the exception of one section, are accessible to anyone with knowledge of elementary calculus; the last four chapters require some knowledge of complex function theory including complex integration and residue calculus.


Язык: en

Рубрика: Математика/Теория чисел/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1976

Количество страниц: 352

Добавлена в каталог: 01.12.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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 $\psi(x)$      75
Chebyshev function $\vartheta(x)$      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 $\chi(n)$      138
Dirichlet, character $\chi(n)$, convolution (product)      29
Dirichlet, character $\chi(n)$, divisor problem      59
Dirichlet, character $\chi(n)$, estimate for d(n)      53 57
Dirichlet, character $\chi(n)$, inverse      30
Dirichlet, character $\chi(n)$, L-function      224
Dirichlet, character $\chi(n)$, series      224
Dirichlet, character $\chi(n)$, 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 $\sigma_{\alpha}(n)$      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 $\varphi(n)$      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 $(-1\mid p)$      181
Evaluation of $(2\mid p)$      181
Evaluation of $L(0,\chi)$      268
Evaluation of $\zeta(-n,a)$      264
Evaluation of $\zeta(2n)$      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, $\pi(x)$      8 74
Function, $\psi(x)$      278
Function, arithmetical      24
Function, Bernoulli periodic $B_n(x)$      267
Function, Chebyshev $\psi(x)$      75
Function, Chebyshev $\vartheta(x)$      75
Function, completely multiplicative      33
Function, Dirichlet $L(s,\chi)$      224
Function, divisor $d(n)$, $\sigma_{\alpha}(n)$      38
Function, Euler totient $\varphi(n)$      25
Function, Hurwitz zeta $\zeta(s,a)$      249
Function, k(n)      247
Function, Liouville $\lambda(n)$      37
Function, M(x)      91
Function, Mangoldt $\Lambda(n)$      32
Function, Mobius $\mu(n)$      24
Function, periodic zeta F(x, s)      257
Function, Riemann zeta $\zeta(s)$      9 249
Function, v(n)      247
Functional equation for $L(s,\chi)$      263
Functional equation for $\zeat(s)$      259
Functional equation for $\zeat(s,h/k)$      261
Functional equation, for $\Gamma(s)$      250
Fundamental Theorem of Arithmetic      17
Gamma function      250
Gauss sum, associated with $\chi$      165
Gauss sum, associated with $\chi$, 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 $\zeta(s,a)$      257
Hurwitz zeta function $\zeta(s,a)$      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 $\mid L(s,\chi)\mid$      1 272
Inequalities for $\mid\zeta(s)\mid$      270 287 291
Inequalities for $\mid\zeta(s,a)\mid$      270
Inequalities for $\pi(n)$      82
Inequalities for $\varphi(n)$      298
Inequalities for d(n)      294
Inequalities for nth prime $p_n$      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 $(n\mid P)$      188
Jacobi triple product identity      319
Jacobi, Carl Gustav Jacob      187 305 313 319
Jordan totient $J_k(n)$      48
Kloosterman sum      176
Kloosterman, H. D.      176
Kolberg, Oddmund      324
Kolesnik, G. A.      59
Kruyswijk, D.      324 327 331
L-function $L(s, \chi)$      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 $(n\mid p)$      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 $\lambda(n)$      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 $\mu(n)$      24
Mobius function $\mu_k(n)$ 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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