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

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Ќазвание: Congruences for L-Functions

јвторы: Urbanowicz J., Williams K.S.

јннотаци€:

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: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2000

 оличество страниц: 269

ƒобавлена в каталог: 01.12.2013

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный 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 55Ч56 62 74Ч75
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 58Ч59 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.      6Ч9 33Ч34 39Ч42 45 60 62 64 72 84 232
Bernoulli numbers      9 11Ч12
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 10Ч12 19 21 28 30 72 232
Boulling, R.      6 232
Brauckman, B.      25 232
Browkin, J.      20 24Ч26 77 81 95Ч96 231Ч233
Brown, E.      55Ч56 58Ч59 61Ч62 74Ч75 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 14Ч15 124 233
Cauchy, A.L.      39 233
Character analogue of the Poisson formula      8 28Ч29 31 34
Chowla, S.      7Ч8 13 39 53 73 231Ч233
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.      56Ч57 59 82 231 233
Coleman formulae      xi 122
Coleman, R.F.      xi 64 122Ч123 125Ч126 132 140Ч141 234
Congruences for $a^{k}(mod 2^{ord_{2}k+6})$      98Ч100
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 111Ч112 114Ч115
Congruences for $k_{2}(-4p)(mod 2, 4, 8)$      95
Congruences for $k_{2}(-8p)(mod 2, 4, 8)$      95Ч96
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 $k_{2}(d)(d&lt;0)$      176 179
Congruences for $k_{2}(d)(d&gt;0)$      78Ч84 86 88 90Ч91 93 104Ч107 111Ч112 114Ч115 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 174Ч177 179
Congruences for $\mathcal{L}_{k, e}(s)$      156 158
Congruences for h(-4p)(mod 4, 8, 16, 128)      55Ч59 65 92 170 172Ч173
Congruences for h(-4pq)(mod 8, 16)      61Ч62 65
Congruences for h(-4pqr)(mod 32)      68Ч69
Congruences for h(-8p)(mod 4, 8, 16, 128)      58Ч60 65 75 92 170 172
Congruences for h(-8pq)(mod 8, 16)      61Ч62 65
Congruences for h(-8pqr)(mod 32)      68Ч69
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)      60Ч61 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)      74Ч75 170 172
Congruences for h(d)      51 76 168Ч172
Congruences for h(d)(d<0)      ix 52 63Ч64 78 80Ч81 86 88 90Ч91 93 97 104Ч107 111Ч112 114Ч115 174 179
Congruences for h(d)(d>0)      53 176Ч177
Congruences for h(p)(mod 4, 8, 16)      53 57Ч58 73 170 172Ч173
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      26Ч27 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 95Ч96 234
Cooke, G.      57 233
Costa, A.      63Ч64 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 2Ч4 16 21Ч22 27Ч28 40 53Ч54 58 72 198 203
Dirichlet L-functions      16 18 119Ч120
Dirichlet regulator      21
Dirichlet, P.G.L.      2Ч3 6 39Ч41 53 55 235Ч236
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 119Ч120 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 $k_{2}(d)(d&lt;0)$      30
Formulae for $k_{2}(d)(d&gt;0)$      29 34 43 78Ч79 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 $\mathcal{L}_{k, \psi}(s)$      127Ч130
Formulae for h(d)      30
Formulae for h(d)(d<0)      2Ч3 5Ч6 27Ч28 33Ч34 39Ч41 43Ч44 79 101 103 183
Formulae for h(d)(d>0)      28
Fox, G.J.      xi 77 203 207Ч208 235
Fresnel, J.      119 231 235
Friesen, C.      74 235
Frobenius polynomials      14 123Ч124
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      55Ч56 58Ч59 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 55Ч56 60 234 236
Gebhardt, H.M.      25 236
Generalized Bernoulli numbers      12Ч13 18 31 70 124 213
Generalized Bernoulli polynomials      13Ч14
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.      133Ч134 181
Gras, G.      xЧxi 64 72 76 97 117 128 134 150 160Ч161 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 Ч Williams congruence      62Ч64 68Ч69 93
Hardy, K.      ix xi 6 49 51 61Ч64 71 75 128 165 167 169 174 236
Hasse classical Klassenzahlbericht      76
Hasse, H.      2 56 58 60 75Ч76 83Ч84 184 237
Hecke, E.      22 27 237
Hikita, M.      76 168 237
Holden, H.      3 6 40Ч41 45 237Ч238
Hudson, R.H.      3 6 44 238
Hurrelbrink, J.      25 27 30 63 76 81 95Ч96 234 238
Hurwitz, A.      13 41 54 238
Ireland, K.      3 10Ч12 16 238
Iwasawa, K.      12 119 238
Jacobi symbols      1
Johnson, W.      3 5 7 45 202 238
K-theorelic background      23
Kaplan, P.      56Ч58 61Ч62 75Ч76 168 172Ч173 236 238Ч239
Karpinski, L.C.      3 41 239
Kenku, M.A.      64 167 239
Keune, F.      24Ч25 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.      25Ч27 81 122 238 240
Kramer, K.      25 81 233
Kronecker symbols      2Ч3
Kronecker, L.      53 240
1 2
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