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http://functions.wolfram.com/10.06.06.0014.01
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LerchPhi[Exp[2 Pi I t], s, p/q] == (2 q Pi)^(s - 1) Gamma[1 - s]
(Sum[Zeta[1 - s, (t + k - 1)/q] Exp[((1 - s) Pi I)/2 -
(2 (t + k - 1) Pi I p)/q], {k, 1, q}] +
Sum[Zeta[1 - s, (k - t)/q] Exp[-(((1 - s) Pi I)/2) +
(2 (k - t) Pi I p)/q], {k, 1, q}]) /;
Element[p, Integers] && p >= 0 && Element[q, Integers] && q > 0 && p <= q
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List[RowBox[List["Exp", "[", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]", " ", "t"]], "]"]], ",", "s", ",", FractionBox["p", "q"]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "q", " ", "\[Pi]"]], ")"]], RowBox[List["s", "-", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "s"]], ",", FractionBox[RowBox[List["t", "+", "k", "-", "1"]], "q"]]], "]"]], RowBox[List["Exp", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "s"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]], "2"], "-", FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["t", "+", "k", "-", "1"]], ")"]], "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"]]], "]"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "s"]], ",", FractionBox[RowBox[List["k", "-", "t"]], "q"]]], "]"]], RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "s"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "+", FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "-", "t"]], ")"]], "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"]]], "]"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["p", ",", "Integers"]], "]"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["q", ",", "Integers"]], "]"]], "\[And]", RowBox[List["q", ">", "0"]], "\[And]", RowBox[List["p", "\[LessEqual]", "q"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> </msup> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "t"]]], Rule[Editable, True]], ",", TagBox["s", Rule[Editable, True]], ",", TagBox[FractionBox["p", "q"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> - </mo> <mi> t </mi> </mrow> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["1", "-", "s"]], Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "-", "t"]], "q"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mi> q </mi> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> t </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["1", "-", "s"]], Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "t", "-", "1"]], "q"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> t </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> q </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> ≤ </mo> <mi> q </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LerchPhi </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> t </ci> </apply> </apply> <ci> s </ci> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> <pi /> </apply> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <times /> <apply> <ci> Zeta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <pi /> <imaginaryi /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <pi /> <imaginaryi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <times /> <apply> <ci> Zeta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> k </ci> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <exp /> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <pi /> <imaginaryi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> k </ci> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> <pi /> <imaginaryi /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> q </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <leq /> <ci> p </ci> <ci> q </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LerchPhi", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "t_"]]], ",", "s_", ",", FractionBox["p_", "q_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "q", " ", "\[Pi]"]], ")"]], RowBox[List["s", "-", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "s"]], ",", FractionBox[RowBox[List["t", "+", "k", "-", "1"]], "q"]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "s"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["t", "+", "k", "-", "1"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "s"]], ",", FractionBox[RowBox[List["k", "-", "t"]], "q"]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "s"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "t"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "0"]], "&&", RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "0"]], "&&", RowBox[List["p", "\[LessEqual]", "q"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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