DirichletCharacter
DirichletCharacter
DirichletCharacter[k,j,n]
给出具有模 k 和指标 j 的狄利克雷特征 
.
更多信息
- 整数型数学函数，同时适合符号和数值操作.
- 对于模 k 的狄利克雷特征，DirichletCharacter[k,j,n] 选择一个特定的排序.
- 对于给定的模 k，有 ϕ
个不同的狄利克雷特征，由指标 j 标记. 对于可能的狄利克雷特征，不同的惯例可能给出不同的排序. - DirichletCharacter[k,j,n] 对 n 是周期性的，并具有周期 k.
- 当 n 和 k 不互质时，DirichletCharacter[k,j,n] 是零.
- DirichletCharacter[k,j,n] 是 n 的可乘函数.
范例
打开所有单元 关闭所有单元基本范例 (2)
Table[DirichletCharacter[2, 1, n], {n, 10}]Table[DirichletCharacter[7, j, n], {j, 1, EulerPhi[7]}, {n, 0, 6}]//GridTable[DiscretePlot[{Re[DirichletCharacter[7, j, n]], Im[DirichletCharacter[7, j, n]]}, {n, 0, 20}, PlotLabel -> j], {j, 1, 6}]范围 (3)
DirichletCharacter[20!, 300, 23]DirichletTransform[DirichletCharacter[3, 2, n], n, s]DirichletCharacter 按元素逐项作用于列表：
DirichletCharacter[3, 2, {1, 2, 3, 4, 5}]应用 (5)
q[k_] := With[{r = Cases[#, {p_, 1} :> p], s = Cases[#, {p_, d_} /; d ≥ 2 :> p]}, k Product[1 - 2 / p, {p, r}]Product[(1 - 1 / p) ^ 2, {p, s}]] & @ FactorInteger[k]Table[q[k], {k, 2, 10}]DiscretePlot[q[k], {k, 2, 50}]用 DirichletCharacter 定义广义伯努利数：
GeneralizedBernoulliB[k_, j_, n_Integer] :=
n! SeriesCoefficient[Series[Sum[DirichletCharacter[k, j, i]t Exp[i * t] / (Exp[k * t] - 1), {i, k - 1}], {t, 0, n + 1}], n]Table[GeneralizedBernoulliB[5, 3, k], {k, 0, 10}]使用广义伯努利数计算 DirichletL 在负整数的值：
SpecialDirichletL[k_, j_, n_Integer ? Negative] := -GeneralizedBernoulliB[k, j, 1 - n] / (1 - n)Table[SpecialDirichletL[5, 3, k], {k, -10, -1}]Table[DirichletL[5, 3, k], {k, -10, -1}]![(TemplateBox[{k}, EulerPhi])/k](Files/DirichletCharacter.zh/1.png "(TemplateBox[{k}, EulerPhi])/k"){kind=small}
Table[GeneralizedBernoulliB[5, j, 0], {j, 4}]
G[k_] := Table[DirichletCharacter[k, j, n], {j, EulerPhi[k]}]add[DirichletCharacter[k_, j1_, n_], DirichletCharacter[k_, j2_, n_]] := DirichletCharacter[k, Mod[j1 + j2, EulerPhi[k], 1], n]e[k_] := DirichletCharacter[k, EulerPhi[k], n]inv[DirichletCharacter[k_, j_, n_]] := DirichletCharacter[k, Mod[EulerPhi[k] - j, EulerPhi[k], 1], n]f = DirichletCharacter[5, 2, n];
g = DirichletCharacter[5, 3, n];add[f, g]{add[f, e[5]], add[e[5], g]}{inv[f], inv[g]}Overscript[χ, ^ ][k_, j_, r_ : 1] := Sum[DirichletCharacter[k, j, i] * Exp[2Pi I r i / k], {i, 1, k - 1}]Overscript[χ, ^ ][7, 4]
Table[DirichletCharacter[7, 4, n]Overscript[χ, ^ ][7, 4, n] == Overscript[χ, ^ ][7, 4], {n, 1, 6}]
Table[Overscript[χ, ^ ][7, 4, 7n], {n, 5}]
FullSimplify[Abs[Overscript[χ, ^ ][7, 4]] == Sqrt[7]]
conductor[k_ , j_] :=
Select[{#, EulerPhi[#] * (j - 1) / EulerPhi[k] + 1} & /@ Divisors[k], (IntegerQ[Last[#]] && Last[#] ≠ 1 && Last[#] ≠ j)&, 1]DirichletCharacter[25,11,n] 有前导子 5：
conductor[25, 11]coprimes = Pick[Range[25], CoprimeQ[Range[25], 25]];Table[DirichletCharacter[25, 11, n] == DirichletCharacter[5, 3, n], {n, coprimes}]属性和关系 (11)
DirichletCharacter 是周期函数：
Table[DirichletCharacter[7, 3, n] == DirichletCharacter[7, 3, 7 + n], {n, 7}]DirichletCharacter 是完全积性的：
DirichletCharacter[7, 3, 4 5]DirichletCharacter[7, 3, 4]DirichletCharacter[7, 3, 5]Table[DirichletCharacter[7, 5, n], {n, 0, 4}]Abs[%]DirichletCharacter 在与 
互质的值处模 
非零：
Table[DirichletCharacter[7, 3, n], {n, 1, 6}]DirichletCharacter 在不与 
互质的值处模 
是零：
Table[DirichletCharacter[7, 3, 7n], {n, 1, 6}]
Table[DirichletCharacter[1, 1, n], {n, -5, 5}]
Table[DirichletCharacter[5, 1, n], {n, EulerPhi[5]}]Table[DirichletCharacter[9, 1, n], {n, EulerPhi[9]}]
Table[DirichletCharacter[k, 1, n], {k, 3, 10}, {n, 0, k - 1}]Table[DirichletCharacter[k, EulerPhi[k] / 2 + 1, n], {k, 3, 10}, {n, 0, k - 1}]对奇数整数 k，JacobiSymbol[n,k] 是实数狄利克雷特征模 k：
Table[JacobiSymbol[n, 5], {n, 0, 4}]Table[DirichletCharacter[5, 3, n], {n, 0, 4}]实数本原特征 χ 模 k 可以定义为 JacobiSymbol[χ[-1]k,n]：
Table[DirichletCharacter[7, 4, n], {n, 0, 6}]Table[JacobiSymbol[DirichletCharacter[7, 4, -1] * 7, n], {n, 0, 6}]在与 
互质的整数处，非原实特征可以用 JacobiSymbol 来表示：
DirichletCharacter[12, 3, {1, 5, 7, 11}]JacobiSymbol[-4, {1, 5, 7, 11}]当 k 的原根 n 存在时，DirichletCharacter[k,j,n] 在该原根处给出 ![exp((2 pi ⅈ (j-1))/(TemplateBox[{k}, EulerPhi]))](Files/DirichletCharacter.zh/4.png "exp((2 pi ⅈ (j-1))/(TemplateBox[{k}, EulerPhi]))"){kind=small}
：
PrimitiveRoot[7]Table[DirichletCharacter[7, j, 3], {j, EulerPhi[7]}]使用 DirichletCharacter 的乘法属性，得到在与 
互质的整数处的值：
Table[DirichletCharacter[7, 2, 3 ^ n] == DirichletCharacter[7, 2, 3] ^ n, {n, EulerPhi[7]}]
DirichletCharacter[3^2 5, 8, 7]DirichletCharacter[3^2, 2, 7]DirichletCharacter[5, 4, 2]p1 = PrimitiveRoot[3^2]p2 = PrimitiveRoot[5]a1 = ChineseRemainder[{p1, 1}, {3^2, 5}]a2 = ChineseRemainder[{1, p2}, {3^2, 5}]e1 = MultiplicativeOrder[a1, 3^2, 7]e2 = MultiplicativeOrder[a2, 5, 7]j1 = 2;j2 = 4;(j1 - 1) * (EulerPhi[3^2] - 1) + (j2 - 1)DirichletCharacter[3^2, j1, a1 ^ e1]DirichletCharacter[5, j2, a2 ^ e2]coprimes = Pick[Range[3^2 5], CoprimeQ[Range[3^25], 3^25]];Table[DirichletCharacter[3^2, j1, a1 ^ MultiplicativeOrder[a1, 3^2, n]]DirichletCharacter[5, j2, a2 ^ MultiplicativeOrder[a2, 5, n]], {n, coprimes}]Table[DirichletCharacter[3^25, 8, n], {n, coprimes}]% == %%Table[DirichletCharacter[3, 2, n], {n, {1, 2, 4, 7, 8, 11, 13, 14}}]index[μ_, k_, j_] := EulerPhi[μ](j - 1) / EulerPhi[k] + 1{index[3^3, 3, 2], index[3^4, 3, 2]}Table[DirichletCharacter[3^3, 10, n], {n, {1, 2, 4, 7, 8, 11, 13, 14}}]Table[DirichletCharacter[3^4, 28, n], {n, {1, 2, 4, 7, 8, 11, 13, 14}}]
Wolfram Research (2008)，DirichletCharacter，Wolfram 语言函数，https://reference.wolfram.com/language/ref/DirichletCharacter.html.
文本
Wolfram Research (2008)，DirichletCharacter，Wolfram 语言函数，https://reference.wolfram.com/language/ref/DirichletCharacter.html.
CMS
Wolfram 语言. 2008. "DirichletCharacter." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/DirichletCharacter.html.
APA
Wolfram 语言. (2008). DirichletCharacter. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/DirichletCharacter.html 年