PolyLog[2, E^((2 Pi I p)/q)] == (1/q^2) Sum[E^((2 Pi I p k)/q) Zeta[2, k/q], {k, 1, q}] /; Element[p, Integers] && p > 0 && Element[q, Integers] && q > 0 && p <= q

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["q", "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p", " ", "k"]], "q"]], RowBox[List["Zeta", "[", RowBox[List["2", ",", FractionBox["k", "q"]]], "]"]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", ">", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", ">", "0"]], "\[And]", RowBox[List["p", "\[LessEqual]", "q"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mi> q </mi> </mfrac> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> q </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mi> q </mi> </mfrac> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mfrac> <mi> k </mi> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;2&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;k&quot;, &quot;q&quot;], Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8804; </mo> <mi> q </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> q </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> p </ci> <ci> k </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <integers /> </apply> <apply> <in /> <ci> q </ci> <integers /> </apply> <apply> <leq /> <ci> p </ci> <ci> q </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p_"]], "q_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p", " ", "k"]], "q"]], " ", RowBox[List["Zeta", "[", RowBox[List["2", ",", FractionBox["k", "q"]]], "]"]]]]]], SuperscriptBox["q", "2"]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", ">", "0"]], "&&", RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "0"]], "&&", RowBox[List["p", "\[LessEqual]", "q"]]]]]]]]]]

2001-10-29