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 FractionalPart

 http://functions.wolfram.com/04.05.21.0011.01

 Input Form

 Integrate[t^(\[Alpha] - 1) FractionalPart[t], {t, 0, Infinity}] == Zeta[-\[Alpha]]/\[Alpha] /; -1 < Re[\[Alpha]] < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["FractionalPart", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List["Zeta", "[", RowBox[List["-", "\[Alpha]"]], "]"]], "\[Alpha]"]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "0"]]]]]]

 MathML Form

 0 t α - 1 frac ( t ) t ζ ( - α ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["-", "\[Alpha]"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] α /; - 1 < Re ( α ) < 0 Condition t 0 t α -1 FractionalPart t Zeta -1 α α -1 -1 α 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["FractionalPart", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["Zeta", "[", RowBox[List["-", "\[Alpha]"]], "]"]], "\[Alpha]"], "/;", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "0"]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29