Urbanowicz J., Williams K.S. — Congruences for L-Functions :: Электронная библиотека попечительского совета мехмата МГУ

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Urbanowicz J., Williams K.S. — Congruences for L-Functions

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Название: Congruences for L-Functions

Авторы: Urbanowicz J., Williams K.S.

Аннотация:

This book provides a comprehensive and up-to-date treatment of research carried out in the last twenty years on congruences involving the values of L-functions (attached to quadratic characters) at certain special values.
There is no other book on the market which deals with this subject. The book presents in a unified way congruences found by many authors over the years, from the classical ones of Gauss and Dirichlet to the recent ones of Gras, Vehara, and others.
Audience: This book is aimed at graduate students and researchers interested in (analytic) number theory, functions of a complex variable and special functions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 269

Добавлена в каталог: 01.12.2013

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Предметный указатель

$(c_{n})$       144
$(t_{n})$       143
$(z_{n})$ , $(u_{n})$       142 151
$A_{5}(d)$ , $A_{6}(d)$       110 111
$A_{i}(d, k)(i = 1, 2, 3, 4)$       104 106 110
$B^{[d]}_{m, \chi}$       70
$B_{n, \chi}$ , nth generalized Bernoulli number      13
$B_{n, \chi}(X)$ , nth generalized Bernoulli polynomial      14
$B_{n}$ , nth Bernoulli number      9
$B_{n}(X)$ , nth Bernoulli polynomial      10
$C = C(\mathcal{D}, q_{1}, q_{2})$       182
$Cl_{2}(F)$       24
$Cl_{2}(t)$ , Clausen function      20
$C_{1}(d)$ , $C_{2}(d)$ , $C_{3}(d)$       101
$c_{\psi} = c_{\psi}(\mathcal{D}, q_{1}, q_{2})$       181
$c_{\psi}$       193
$D_{n}$ , nth D-number      13
$E_{n}$ , nth Euler number      13
$F\pm(X)$       54
$F^{*}\wedge F^{*}$ , modified external product      25
$F^{+}$ , maximal totally real subfield of F      23
$g_{2r+\varrho}$ , $v_{2r+1}$       159
$G_{e}(x)$       139
$g_{P}$       184
$H(d, \mathcal{D})$       63
$H(\mathcal{D})$ , $K_{2}(\mathcal{D})$       166
$h^{+}$ , $h^{-}$       23
$h_{0}(d)$ , number of narrow classes of $\mathbb{Q}(\sqrt{d})$       30
$H_{1}(d)$ , $H_{2}(d)$       63
$H_{k}(d)$       113
$K(d, \mathcal{D})$       93
$K_{0}$ , $K_{1}$ , $K_{2}$ , Milnor K-functors      24
$k_{2}(d)$ , order of the group $K_{2}O_{F}$ for $F = \mathbb{Q}(\sqrt{d})$       29
$K_{i}(d)(i = 1, 2, 3, 4, 5)$       94
$K_{n}(n\geq 0)$ , Quillen K-functors      23
$L(s, \chi)$ , Dirichlet L-function      16
$Li_{2}(s)$ , Euler dilogarithm      19
$log_{p}s = log_{p}(s)$ , p-adic logarithm      118
$l^{(p)}_{k}(s)$       126
$L^{+}(s)$ , $L^{-}(s)$ , $K^{\pm}_{e}$       127
$L^{[m, \Theta]}_{2}(k, \chi_{ed}\omega^{1-k})$       160
$l_{k}(s) = l_{k, p}(s)$ , kth p-adic multilogarithm      122
$l_{k}(s) = l_{k, \infty}(s)$ , kth complex multilogarithm      122
$L_{p}(s, \chi)$ , p-adic L-function      119
$N\mathfrak{a} = N(\mathfrak{a})$       21
$O_{F}$ , ring of integers of a number field F      21
$r_{1}$ , $r_{2}$       21
$R_{2, p} = R_{2, p}(d)$ , second p-adic regulator of $\mathbb{Q}(\sqrt{d})$       122
$r_{2}(A)$ , Sylow 2-rank of a finite abelian group A      24
$R_{2}(F)$ , second Borel regulator of F      26
$R_{n}(u)$ , nth Frobenius polynomial      14
$R_{p} = R_{p}(F)$ , p-adic regulator of F      121
$S(\mathcal{D}, q_{1}, q_{2})$       181
$S^{*}_{m}(N/8, \chi)$       36
$S_{k, \chi}(T)$       129
$S_{m}(N, \chi)$       13
$s_{m}(r, k, \chi)$       33
$S_{m}(x, r, s, \chi)$       45
$S_{m}(x, \chi)$       36
$S_{n}(x)$       10
$T_{i}$ , (i = 1, 2, 3, 4)      108
$w_{2}(d)$ , $w_{2}(F)$ for $F = \mathbb{Q}(\sqrt{d})$       29
$W_{k, e}(n)$       134
$w_{r} = w_{r}(F)$ , largest integer s such that the group $G(F(\zeta_{s})/F)$ is annihilated by r      26
$w_{\alpha} = w_{\alpha}(\xi)$       137
$x = \{x_{k, e}\}$       141
$y_{n}(x)(n\geq 0)$       142
$\chi_{1}$ , trivial primitive character      2
$\chi_{d}(n) = \left(\frac{d}{n}\right)$ , Kronecker symbol      2
$\epsilon$ , $\rho$       97
$\Gamma(s)$ , Gamma function      11
$\gamma_{n, e}$       134
$\gamma_{n}$       133
$\Lambda$ , $\Lambda_{-1}$ , $\Lambda_{0}$ , $\Lambda_{1}$ , $\Lambda_{2}$ , $\Lambda^{'}_{-1}$ , $\Lambda^{'}_{1}$       166
$\Lambda_{1}(m)$ , $\Lambda(x, m, \Psi)$ , $\Lambda(x, m, \Psi, \Theta)$       161 164
$\Lambda_{k, \psi} = \Lambda_{k, \psi}(N, \chi)$       129
$\left(X\atop n\right)$       120
$\left(\frac{m}{n}\right)_{4}$ , biquadratic symbol      56 57 59
$\left(\frac{n}{m}\right)$ , Jacobi symbol      1
$\left(\frac{n}{p}\right)$ , Legendre symbol      1
$\mathcal{A}_{k, \chi}(R)$       37
$\mathcal{B}(k, e, d)$       229
$\mathcal{F}$       181
$\mathcal{L}_{k, e}(\xi)$       137
$\mathcal{L}_{k, \psi}(s)$       127
$\mathcal{L}_{\chi}(s)$       31
$\mathcal{P}_{0}$ , $\mathcal{P}_{1}$       182
$\mathcal{R}_{d}$ , $\mathcal{R}^{+}_{d}$ , $\mathcal{R}^{-}_{d}$       97
$\mathcal{T}_{r}$ , set of all fundamental discriminants dividing r      69
$\nu_{k, e}$       139
$\omega = \omega_{p}$ , Teichmueller character at p      119
$\tau(\chi)$ , normalized Gauss sum      9
$\tau(\chi, \zeta)$ , Gauss sum      9
$\varepsilon_{d}$ , $\eta_{d}$ , $t_{0}$ , $u_{0}$ , t, u, T, U, $\mu$       171
$\varepsilon_{m, \chi}$       36
$\xi(d)$ , $\eta(d)$       101
$\zeta(s)$ , Riemann zeta function      11
$\zeta_{F, p}(s)$ , p-adic zeta function      121
$\zeta_{F}(s)$ , Dedekind zeta function      21
${\sum\limits^{c}_{a=1}}^{'}$ , sum taken over integers a prime to c      127
${\sum\limits_{a\leq k\leq b}}^{*}$ , if a or b is an integer, then the associated summands are halved      8
<a>      119
A(F), C(F), $\mathbb{D}(C(F))$       25
A(t, n), number of positive integers $\leq t$ prime to n      48
Akiyama, S.      32 231
Ambiguous forms and classes      30 51 55—56 62 74—75
Amice, Y.      119 231
Ankeny, N.C.      13 231
Apostol, T.M.      8 11 231
Ars Conjectandi      10
Artin, E.      13 231
Ayoub, R.      39 231
Barkan, P.      55 58—59 231
Barner, K.      27 231
Barrucand, P.      56 59 82 231
Bauer, H.      75 231
Belabas, K.      25 231
Berger, A.      13 231
Berger, R.I.      25 232
Berndt, B.C.      6—9 33—34 39—42 45 60 62 64 72 84 232
Bernoulli numbers      9 11—12
Bernoulli polynomials      10
Bernoulli, J.      10
Binomial coefficient identities      133 135 151
Birch, B.J.      26 232
Bloch, S.      20 26 232
Boldy, M.C.      25 232
Borel, A.      24 26 232
Borevich, Z.I.      3 10—12 19 21 28 30 72 232
Boulling, R.      6 232
Brauckman, B.      25 232
Browkin, J.      20 24—26 77 81 95—96 231—233
Brown, E.      55—56 58—59 61—62 74—75 82 84 233
c = c(y), exponent of $y = (y_{n})_{n\geq 0}$       142
c(L)      143
Candiotti, A.      25 81 233
Carlitz, L.      10 14—15 124 233
Cauchy, A.L.      39 233
Character analogue of the Poisson formula      8 28—29 31 34
Chowla, S.      7—8 13 39 53 73 231—233
Cl(F), group of ideal classes of a number field F      24
Class number formulae      21 121
Clausen function      20
Clausen, T.      10 20
CM-fields      23
Coates, J.      27 119 233
Cohn, H.      56—57 59 82 231 233
Coleman formulae      xi 122
Coleman, R.F.      xi 64 122—123 125—126 132 140—141 234
Congruences for $a^{k}(mod 2^{ord_{2}k+6})$       98—100

Congruences for $B^{[T]}_{k, \chi}(N)/k$       217 220
Congruences for $B_{k, \chi}/k$       69 77 224
Congruences for $B_{k, \chi}/k(mod 64)$       97 108 111—112 114—115
Congruences for $k_{2}(-4p)(mod 2, 4, 8)$       95
Congruences for $k_{2}(-8p)(mod 2, 4, 8)$       95—96
Congruences for $k_{2}(-p)(mod 2, 4, 8)$       95
Congruences for $k_{2}(-pq)(mod 4, 8, 16)$       96
Congruences for $k_{2}(4p)(mod 8, 32)$       85
Congruences for $k_{2}(8p)(mod 8, 16, 32, 128)$       85 92
Congruences for $k_{2}(d)$       77 175
Congruences for $k_{2}(d)(d<0)$       176 179
Congruences for $k_{2}(d)(d>0)$       78—84 86 88 90—91 93 104—107 111—112 114—115 174 177 210
Congruences for $k_{2}(p)(mod 4, 8, 16, 32, 128)$       85 92
Congruences for $k_{2}(pq)(mod 16, 32)$       85
Congruences for $L(k, \chi)(k\leq 0)$       69
Congruences for $L_{2}(k, \chi\omega^{1-k})$       128 160 163
Congruences for $L_{2}(k, \chi\omega^{1-k})(k = -1, 0, 1, 2)$       161 166 168 174—177 179
Congruences for $\mathcal{L}_{k, e}(s)$       156 158
Congruences for h(-4p)(mod 4, 8, 16, 128)      55—59 65 92 170 172—173
Congruences for h(-4pq)(mod 8, 16)      61—62 65
Congruences for h(-4pqr)(mod 32)      68—69
Congruences for h(-8p)(mod 4, 8, 16, 128)      58—60 65 75 92 170 172
Congruences for h(-8pq)(mod 8, 16)      61—62 65
Congruences for h(-8pqr)(mod 32)      68—69
Congruences for h(-p)(mod 4, 8, 16, 128)      52 54 59 65 74 92 170 172
Congruences for h(-pq)(mod 4, 8, 16)      60—61 65
Congruences for h(-pqr)(mod 8, 32)      68
Congruences for h(4p)(mod 4, 8, 16)      54 74 170 172
Congruences for h(8p)(mod 4, 8, 16)      74—75 170 172
Congruences for h(d)      51 76 168—172
Congruences for h(d)(d<0)      ix 52 63—64 78 80—81 86 88 90—91 93 97 104—107 111—112 114—115 174 179
Congruences for h(d)(d>0)      53 176—177
Congruences for h(p)(mod 4, 8, 16)      53 57—58 73 170 172—173
Congruences for h(pq)(mod 4, 8)      75
Congruences for h(pqr), h(4pq) and h(8pq)      75
Congruences for p-adic numbers      118
Congruences for power sums of consecutive natural numbers      222
Conjecture of Birch and Tate      26—27 29 78 81
Conjecture of Federer      27
Conjecture of Leopoldt      121
Conjecture of Lichtenbaum      26 30 165
Conner, P.E.      25 27 30 76 81 95—96 234
Cooke, G.      57 233
Costa, A.      63—64 234
Currie, J.      8 45 60 234
Cuspidal behaviour of 2-adic modular forms      64
D(s), dilogarithm of Wigner and Bloch      20
d, fundamental discriminant      2
D-numbers      13
Damey, P.      75 234
Davenport, H.      2 184 234
Dedekind zeta function      21
Dedekind, R.      41 55 234
Deligne, P.      119 234
Desnoux, P.-J.      76 167 234
Dilcher, K.      10 234
Dilogarithm of Euler      19
Dilogarithm of Rogers      20 125
Dilogarithm of Wigner and Bloch      20 26
Dirichlet characters      2 184
Dirichlet class number formulae      x 2—4 16 21—22 27—28 40 53—54 58 72 198 203
Dirichlet L-functions      16 18 119—120
Dirichlet regulator      21
Dirichlet, P.G.L.      2—3 6 39—41 53 55 235—236
E = E(F), group of units in F      23
E*(x), R(x), R*(x)      34
Endo, A.      76 235
Ernvall, R.      13 15 235
Euler criterion      1
Euler factors      16 119—120 123
Euler formula for $\zeta(2k)$       12
Euler numbers      13 115
Euler product      16
Euler, L.      12 19 235
Exact hexagon      76
Federer, L.J.      27 235
Fleckinger, V.      26 122 240
Formulae for $B_{1, \chi}$       28 43
Formulae for $B_{2, \chi}$       29
Formulae for $B_{k, \chi}$       100
Formulae for $k_{2}(d)(d<0)$       30
Formulae for $k_{2}(d)(d>0)$       29 34 43 78—79 97 101 104 210
Formulae for $L_{2}(k, \chi\omega^{1-k})$       129
Formulae for $l_{k}(s)$       125
Formulae for $\mathcal{L}_{k, e}(s)$       141
Formulae for $\mathcal{L}_{k, e}(s)(k = -1, 0, 1, 2)$       137
Formulae for $\mathcal{L}_{k, \psi}(s)$       127—130
Formulae for h(d)      30
Formulae for h(d)(d<0)      2—3 5—6 27—28 33—34 39—41 43—44 79 101 103 183
Formulae for h(d)(d>0)      28
Fox, G.J.      xi 77 203 207—208 235
Fresnel, J.      119 231 235
Friesen, C.      74 235
Frobenius polynomials      14 123—124
Frobenius, F.G.      14 122 235
Functional equation for $L(s, \chi)$       16
Functional equation for $\zeta(s)$       11
Functional equation for $\zeta_{F}(s)$       22
Fundamental discriminants      2
g(2)      24
Gamma function      11
Gangl, H.      25 231 233
Garland, H.      24 235
Gauss congruence      55—56 58—59 75 203 208
Gauss evaluation of $\tau (\chi)$       17
Gauss sum      9 17
Gauss theory of ambiguous classes      30 51 55 76
Gauss, C.F.      6 41 55—56 60 234 236
Gebhardt, H.M.      25 236
Generalized Bernoulli numbers      12—13 18 31 70 124 213
Generalized Bernoulli polynomials      13—14
Generalized Kummer congruences      15 97 119
Glaisher, J.W.L.      41 55 236
Goren, E.Z.      97 236
Gradshteyn, I.S.      36 236
Granville, A.      133—134 181
Gras, G.      x—xi 64 72 76 97 117 128 134 150 160—161 165 167 169 175 230 236
Greither, C.      27 236
h = h(F), class number of F      23
h(d), class number of $\mathbb{Q}(\sqrt{d})$       2
Halter-Koch, F.      76 236
Hardy — Williams congruence      62—64 68—69 93
Hardy, K.      ix xi 6 49 51 61—64 71 75 128 165 167 169 174 236
Hasse classical Klassenzahlbericht      76
Hasse, H.      2 56 58 60 75—76 83—84 184 237
Hecke, E.      22 27 237
Hikita, M.      76 168 237
Holden, H.      3 6 40—41 45 237—238
Hudson, R.H.      3 6 44 238
Hurrelbrink, J.      25 27 30 63 76 81 95—96 234 238
Hurwitz, A.      13 41 54 238
Ireland, K.      3 10—12 16 238
Iwasawa, K.      12 119 238
Jacobi symbols      1
Johnson, W.      3 5 7 45 202 238
K-theorelic background      23
Kaplan, P.      56—58 61—62 75—76 168 172—173 236 238—239
Karpinski, L.C.      3 41 239
Kenku, M.A.      64 167 239
Keune, F.      24—25 239
Kisilevsky, H.      63 239
Kleboth, H.      13 239
Klingen, H.      27 239
Koblitz, N.      11 117 239
Koch, H.      76 239
Kohno, Y.      76 240
Kolster, M.      25—27 81 122 238 240
Kramer, K.      25 81 233
Kronecker symbols      2—3
Kronecker, L.      53 240

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