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Apostol T.M. — Modular Functions and Dirichlet Series in Number Theory
Apostol T.M. — Modular Functions and Dirichlet Series in Number Theory



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Название: Modular Functions and Dirichlet Series in Number Theory

Автор: Apostol T.M.

Аннотация:

This volume is a sequel to the author's Introduction to Analytic Number Theory (UTM 1976, 3rd Printing 1986). It presupposes an undergraduate background in number theory comparable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis. Most of this book is devoted to a classical treatment of elliptic and modular functions with some of their number-theoretic applications. Among the major topics covered are Rademacher's convergent series for the partition modular function, Lehner's congruences for the Fourier coefficients of the modular function j, and Hecke's theory of entire forms with multiplicative Fourier coefficients. The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series. In addition to the correction of misprints, minor changes in the exercises and an updated bibliography, this new edition includes an alternative treatment of the transformation formula for the Dedekind eta function, which appears as a five-page supplement to Chapter 3.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 1990

Количество страниц: 204

Добавлена в каталог: 08.07.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$e_1$, $e_2$, $e_3$      13
$g_2$, $g_3$      12
$j(\tau)$, $J(\tau)$      74 15
$j(\tau)$, $J(\tau)$, Fourier coefficients of      21
$\wp$-function of Weierstrass      10
Abscissa, of absolute convergence      165
Abscissa, of convergence      165
Additive number theory      1
Apostol, Tom M.      196
Approximation theorem of Dirichlet      143
Approximation theorem of Kronecker      148 150 154
Approximation theorem of Liouville      146
Asymptotic formula for p(n)      94 104
Atkin, A.O.L.      91 196
Automorphic function      79
Basis for sequence of exponents      166
Bernoulli numbers      132
Bernoulli polynomials      54
Berwick, W.E.H.      22
Bessel functions      109
Bohr function of a Dirichlet series      168
Bohr matrix      167
Bohr, equivalence theorem      178
Bohr, Harald      161 196
Circle method      96
Class number of quadratic form      45
Congruence properties of Dedekind sums      64
Congruence properties, of coefficients of $j(\tau)$      22 90
Congruence subgroup      75
Cusp form      114
Davenport, Harold      136
Dedekind function $\eta(\tau)$      47
Dedekind sums      52 61
Dedekind, Richard      47
Deligne, Pierre      136 140 196
Differential equation for $\wp(z)$      11
Dirichlet L-function      184
Dirichlet series      161
Dirichlet, Peter Gustav Lejeune      143
Dirichlet’s approximation theorem      143
Discriminant $\Delta(\tau)$      14
Divisor functions $\sigma_{\alpha}(n)$      20
Doubly periodic functions      2
Eigenvalues of Hecke operators      129
Eisenstein series $G_n$      12
Eisenstein series $G_n$, recursion formula for      13
Elliptic functions      4
Entire modular forms      114
Equivalence of general Dirichlet series      173
Equivalence of ordinary Dirichlet series      174
Equivalence of pairs of periods      4
Equivalence of points in the upper half-plane H      30
Equivalence of quadratic forms      45
Estimates for coefficients of modular forms      134
Euler products of Dirichlet series      136
Euler, Leonhard      94
Exponents of a general Dirichlet series      161
Farey fractions      98
Ford circles      99
Ford, L.R.      99 196
Fourier coefficients of $j(\tau)$      21 74
Fourier coefficients of $j(\tau)$, divisibility properties of      22 74 91
Functional equation, for $\eta(\tau)$      48 52
Functional equation, for $\Lambda(\alpha, \beta, z)$      54
Functional equation, for $\Phi(\alpha, \beta, s)$      56 71
Functional equation, for $\vartheta(\tau)$      91
Functional equation, for $\zeta(s)$      140
Fundamental pairs of periods      2
Fundamental region of subgroup $\Gamma_0(p)$      76
Fundamental region, of modular group $\Gamma$      31
General Dirichlet series      161
Generators of congruence subgroup $\Gamma_0(p)$      78
Generators, of modular group $\Gamma$      28
Grosswald, Emil      61 198
Gupta, Hansraj      111 196
Half-plane H      14
Half-plane of absolute convergence      165
Half-plane of convergence      165
Hardy — Ramanujan formula for p(n)      94
Hardy, Godfrey Harold      94 196
Haselgrove, C.B.      186 196
Hecke operators $T_n$      120
Hecke, Erich      114 120 133 196 197
Helly selection principle      179
Helly, Eduard      179
Hurwitz approximation theorem      145
Hurwitz zeta function      55 71
Hurwitz, Adolf      55 145
Invariants $g_2$, $g_3$      12
Inversion problem for Eisenstein series      42
Iseki, Sho      52 197
Iseki’s transformation formula      53
Jacobi theta function      91 141
Jacobi triple product identity      91
Jacobi, Carl Gustav Jacob      6 91 141
Klein modular invariant $J(\tau)$      15
Klein, Felix      15
Kloosterman, H.D.      136
Knopp, Marvin I.      197
Kronecker approximation theorem      148 150 154
Kronecker, Leopold      148
Lambert series      24
Lambert, Johann Heinrich      24
Landau, Edmund      186
Lehmer conjecture      22
Lehmer, Derrick Henry      22 93 95 197
Lehner, Joseph      22 91 111 197
LeVeque, William Judson      197
Linear space $M_k$ of entire forms      118
Linear subspace $M_{k,0}$ of cusp forms      119
Liouville approximation theorem      146
Liouville function $\lambda(n)$      25 184
Liouville numbers      147
Liouville, Joseph      5 146 184
Littlewood, John Edensor      95
Mapping properties of $J(\tau)$      40
Mediant      98
Mellin inversion formula      54
Mellin, Robert Hjalmar      54
Mobius function      24 187
Mobius transformation      27
Mobius, Augustus Ferdinand      24 27 187
Modular forms      114
Modular forms and Dirichlet series      136
Modular function      34
Modular group $\Gamma$      28
Modular group $\Gamma$, subgroups of      46 75
Montgomery, H.L.      187
Mordell, Louis Joel      92 197
Multiplicative property of Hecke operators      126 127
Multiplicative property of Ramanujan tau function      93 114
Multiplicative property, of coefficients of entire forms      130
Neville, Eric Harold      110 197
Newman, Morris      91 111 197
Normalized eigenform      130
Order of an elliptic function      6
O’Brien, J.N.      91 196
Partition function p(n)      1 94
Period      1
Period parallelogram      2
Periodic zeta function      55
Petersson inner product      133
Petersson — Ramanujan conjecture      140
Petersson, Hans      22 133 140 197
Picard, Charles Emile      43
Picard’s Theorem      43
Product representation for $\Delta(\tau)$      51
Quadratic forms      45
Rademacher path of integration      102
Rademacher series for p(n)      104
Rademacher, Hans      22 62 95 102 104 197
Ramanujan conjecture      136
Ramanujan tau function      20 22 92 113 131 198
Ramanujan, Srinivasa      20 92 94 136 191
Rankin, Robert A.      136 198
Reciprocity law for Dedekind sums      62
Representative of quadratic form      45
Riemann zeta function      20 140 155 185 189
Riemann, Georg Friedrich Bernhard      140 155 185 198
Rouche, Eugene      180
Rouche’s Theorem      180
Salie, Hans      136
Schoeneberg, Bruno      198
Sczech, R.      61 198
Selberg, Atle      136 198
Serre, Jean-Pierre      198
Siegel, Carl Ludwig      48 198
Simultaneous eigenforms      130
Spitzenform      114
Subgroups of the modular groups      46 75
Tau function      20 22 92 113 131
Theta function      91 141
Transcendental numbers      147
Transformation formula, of Dedekind      48 52
Transformation formula, of Dedekind, of Iseki      54
Transformation of order n      122
Turan, Paul      185 198
Turan’s Theorem      185 186
Univalent modular function      84
Uspensky, J.V.      94 198
Valence of a modular function      84
Values of $J(\tau)$      39
Values of Dirichlet series      170
van Wijngaarden, A.      22
Vertices of fundamental region      34
Watson, G.N.      109 198
Weierstrass $\wp$-function      10
Weierstrass, Karl      6
Weight formula for zeros of an entire form      115
Weight of a modular form      114
Whiteman, Albert Leon      62 198
Zeros, of an elliptic function      5
Zeta function, Hurwitz      55
Zeta function, periodic      55
Zeta function, Riemann      140 155 185 189
Zuckerman, Herbert, S.      22
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