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Levit Department of Computer Systems Center for Technological Education Tel-Aviv University 52 Golomb St., P.O.Box 305 Holon 58102, Israel e-mail: levitv@barley.cteh.ac.il \ \>", "Subtitle", TextAlignment->Center, TextJustification->0, FontSize->13, FontSlant->"Italic"], Cell[CellGroupData[{ Cell["Abstract", "Section", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["The method of the analytic continuation of the function given on \ a part of the boundary into the whole domain is applied to the Riemann \ hypothesis on the Zeta function zeros. Comprehensive numerical experiments \ have been performed with the ", FontSize->12], StyleBox["Mathematica", FontSize->12], StyleBox[".The computational approach confirmed the Riemann hypothesis.", FontSize->12] }], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Section", CellMargins->{{16, 19}, {Inherited, Inherited}}, PageBreakAbove->True], Cell[TextData[{ StyleBox["The famous Riemann zeta function ", FontSize->12], Cell[BoxData[ FormBox[ StyleBox[\(\[Zeta](z)\), FontSize->12], TraditionalForm]]], StyleBox[" is defined by", FontSize->12] }], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ StyleBox[\(\[Zeta]( z) = \[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\^\(\(\ \)\(z\)\)\), FontSize->12], TraditionalForm]], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}, TextAlignment->Center, FontSize->12], Cell[TextData[{ StyleBox["for ", FontSize->12], Cell[BoxData[ \(TraditionalForm\`Re(z) > 1\)]], StyleBox[", and by", FontSize->12] }], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ TagBox[ RowBox[{"\[Zeta]", "(", TagBox[\(1 - z\), (Editable -> True)], ")"}], InterpretTemplate[ Zeta[ #]&]], "=", RowBox[{\(2\^\(1 - z\)\), " ", \(\[Pi]\^\(-z\)\), " ", \(cos(\(\[Pi]\ z\)\/2)\), " ", \(\[CapitalGamma](z)\), " ", TagBox[ RowBox[{"\[Zeta]", "(", TagBox["z", (Editable -> True)], ")"}], InterpretTemplate[ Zeta[ #]&]]}]}], FontSize->12, FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], TraditionalForm]], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}, TextAlignment->Center, FontSize->12], Cell[TextData[{ StyleBox["for ", FontSize->12], Cell[BoxData[ \(TraditionalForm\`Re(z) < 1\)]], StyleBox[". The Riemann zeta function is meromorphic everywhere in the \ complex ", FontSize->12], StyleBox["z", FontSize->12, FontSlant->"Italic"], StyleBox["-plane except for a simple pole at ", FontSize->12], StyleBox["z", FontSize->12, FontSlant->"Italic"], StyleBox[" = 1. In 1859 Riemann, studying the problem of the distribution \ of prime numbers came up with the following ", FontSize->12] }], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["CONJECTURE. ", FontSize->12, FontWeight->"Bold"], StyleBox["All complex zeros of", FontSize->12, FontSlant->"Italic"], StyleBox[" \[Zeta](", FontSize->12], StyleBox["z", FontSize->12, FontSlant->"Italic"], StyleBox[") ", FontSize->12], StyleBox["lie on the critical line", FontSize->12, FontSlant->"Italic"], StyleBox[" ", FontSize->12], Cell[BoxData[ \(TraditionalForm\`Re(z) = 1/2\)]] }], "Text", CellMargins->{{Inherited, 19}, {Inherited, Inherited}}], Cell[TextData[StyleBox["\tThough the great deal of efforts have been \ performed and many outstanding mathematicians have worked on it, the \ hypothesis neither been proved nor disproved. However, the latest numerical \ computations of complex zeros of the Riemann zeta function (see [4],[5] and \ [6]) allow us believe the hypothesis to be true.", FontSize->12]], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[TextData[StyleBox["\tIn this article we shortly describe a new method of \ approaching the Riemann hypothesis. The idea is based on the solution of the \ Carleman boundary value problem, which were successfuly solved by L. \ Aizenberg (see [1], [2] and [3]). It turns out that the Riemann hypothesis \ can be envisaged as the Carleman problem. Armed with the new mathematical \ technique we performed here the computational approach confirmed the Riemann \ hypothesis. It is worth to mention that in comparison with the previous \ numerical verifications of the Riemann hypothesis using supercomputer tools, \ we arrived at our conclusion with only a regular Pentium computer at hand.", FontSize->12]], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Underlying Mathematics", "Section", CellMargins->{{16, 19}, {Inherited, Inherited}}, PageBreakAbove->True], Cell[TextData[{ StyleBox["Let ", FontSize->12], StyleBox["K", FontSize->12, FontSlant->"Italic"], StyleBox[" = {", FontSize->12], StyleBox["z", FontSize->12, FontSlant->"Italic"], StyleBox[" : |", FontSize->12], StyleBox["z", FontSize->12, FontSlant->"Italic"], StyleBox["| < 1} be a unit disk, ", FontSize->12], StyleBox["G", FontSize->12, FontSlant->"Italic"], StyleBox[" be a chord, connecting two points of \[PartialD]K and ", FontSize->12], StyleBox["D", FontSize->12, FontSlant->"Italic"], StyleBox[" be a segment between ", FontSize->12], StyleBox["G", 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Consider \ the strip ", StyleBox["K", FontSlant->"Italic"], " = { ", StyleBox["z", FontSlant->"Italic"], " : 0 < ", Cell[BoxData[ \(TraditionalForm\`Re(z)\)]], StyleBox[" ", FontSlant->"Italic"], "< ", Cell[BoxData[ \(TraditionalForm\`1\)]], "} and the function", Cell[BoxData[ FormBox[ StyleBox[\(1\/\(\[Zeta](z)\)\), FontSize->16], TraditionalForm]]], " defined on the interval (0,", Cell[BoxData[ \(TraditionalForm\`1\)]], "). Our goal is to analytically continue the function", Cell[BoxData[ FormBox[ StyleBox[\(1\/\(\[Zeta](z)\)\), FontSize->16], TraditionalForm]]], " from the given interval into the whole strip ", StyleBox["K", FontSlant->"Italic"], ". Taking into account that all complex zeros of the zeta function lie in \ the strip ", StyleBox["K", FontSlant->"Italic"], " and they are symmetrical so that if \[Rho] is a zero then ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&_\)]], ", 1-\[Rho] and 1-", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&_\)]], " are also zeros, it is quite sufficient to consider the half strip ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega]\)]], " = { ", StyleBox["z", FontSlant->"Italic"], " : 0 < ", Cell[BoxData[ \(TraditionalForm\`Re(z)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "} and find out whether ", Cell[BoxData[ FormBox[ StyleBox[\(1\/\(\[Zeta](z)\)\), FontSize->16], TraditionalForm]]], " can be analytically continued into ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega]\)]], ". If that is a case then it would mean that ", Cell[BoxData[ FormBox[ StyleBox[\(1\/\(\[Zeta](z)\)\), FontSize->16], TraditionalForm]]], "does not have any singular points in ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega]\)]], ", or, other words, that there are no zeros of the zeta function in ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega]\)]], ", and thus in ", StyleBox["K. ", FontSlant->"Italic"], "It would follow then that all zeros are on a critical line ", Cell[BoxData[ \(TraditionalForm\`Re(z)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "." }], "Text", CellMargins->{{14, 19}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ "\tTo make a use of the necessary and sufficient conditions decribed in the \ previous section, we have to map the strip ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega]\)]], " into the sector of the unit disk. 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