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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->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[ " Version 3.0. The computational approach confirmed the Riemann \ hypothesis.", FontSize->14] }], "Abstract", CellMargins->{{16, 19}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Section", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["The famous Riemann zeta function ", FontSize->14], Cell[BoxData[ FormBox[ StyleBox[\(\[Zeta](z)\), FontSize->14], TraditionalForm]]], StyleBox[" is defined by", FontSize->14] }], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ StyleBox[\(\[Zeta](z) = \[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\^\(\ z\)\), FontSize->14], TraditionalForm]], "Input", CellMargins->{{16, 19}, {Inherited, Inherited}}, TextAlignment->Center, FontSize->12], Cell[TextData[{ StyleBox["for ", FontSize->14], StyleBox["\[GothicCapitalR]\[GothicE]", FontFamily->"Times", FontSize->14, FontWeight->"Roman", FontSlant->"Italic", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["(", FontSize->14], StyleBox["z)", FontSize->14, FontSlant->"Italic"], StyleBox[" > 1, and by", FontSize->14] }], "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[ #]&]]}]}], FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], TraditionalForm]], "Input", CellMargins->{{16, 19}, {Inherited, Inherited}}, TextAlignment->Center, FontSize->12], Cell[TextData[{ StyleBox["for ", FontSize->14], StyleBox["\[GothicCapitalR]\[GothicE]", FontFamily->"Times", FontSize->14, FontWeight->"Roman", FontSlant->"Italic", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["(", FontSize->14], StyleBox["z)", FontSize->14, FontSlant->"Italic"], StyleBox[ " < 1. The Riemann zeta function is meromorphic everywhere in the complex ", FontSize->14], StyleBox["z", FontSize->14, FontSlant->"Italic"], StyleBox["-plane except for a simple pole at ", FontSize->14], StyleBox["z", FontSize->14, FontSlant->"Italic"], StyleBox[" = 1. In 1859 Riemann ", FontSize->14], StyleBox["studying", FontSize->16], StyleBox[ " the problem of the distribution of prime numbers came up with the \ following ", FontSize->14] }], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[TextData[StyleBox["Conjecture", FontSize->14, FontWeight->"Bold"]], "Text", CellMargins->{{Inherited, 19}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["\t", FontSize->14], StyleBox["All complex zeros of", FontSize->14, FontSlant->"Italic"], StyleBox[" \[Zeta](", FontSize->14], StyleBox["z", FontSize->14, FontSlant->"Italic"], StyleBox[") ", FontSize->14], StyleBox["lie on the critical line", FontSize->14, FontSlant->"Italic"], StyleBox[" ", FontSize->14], StyleBox["\[GothicCapitalR]\[GothicE]", FontFamily->"Times", FontSize->14, FontWeight->"Roman", FontSlant->"Italic", FontTracking->"Plain", PrivateFontOptions->{"FontPostScriptName"->Automatic}], StyleBox["(", FontSize->14], StyleBox["z", FontSize->14], StyleBox[")", FontSize->14, FontSlant->"Italic"], StyleBox[" = ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`1\/2\)]] }], "Text", CellMargins->{{16, 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->14]], "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->14]], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Underlying Mathematics", "Section", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["Let ", FontSize->14], StyleBox["K", FontSize->14, FontSlant->"Italic"], StyleBox[" = {", FontSize->14], StyleBox["z", FontSize->14, FontSlant->"Italic"], StyleBox[" : |", FontSize->14], StyleBox["z", FontSize->14, FontSlant->"Italic"], StyleBox["| < 1} be a unit disk, ", FontSize->14], StyleBox["G", FontSize->14, FontSlant->"Italic"], StyleBox[" be a chord, connecting two points of \[PartialD]K and ", FontSize->14], StyleBox["D", 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The Carleman problem is formulated as it follows: ", FontSize->14], StyleBox["whether there exists an analytic continuation of function ", FontSize->14, FontSlant->"Italic"], StyleBox["f", FontSize->14, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" given on a boundary G into domain D.", FontSize->14, FontSlant->"Italic"], StyleBox[" In has been proved (see [Aiz,1995]) that the condition", FontSize->14] }], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ OverscriptBox[ UnderscriptBox["lim", RowBox[{"n", "\[Rule]", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}]], "_"], "\[ThinSpace]", RadicalBox[\(\[LeftBracketingBar]a\_n\[RightBracketingBar]\), StyleBox["n", FontSize->12.5625, FontSlant->"Italic"]]}], " ", "=", " ", "1"}], ","}], FontSize->14], TraditionalForm]], "Input", CellMargins->{{Inherited, 19}, {Inherited, Inherited}}, TextAlignment->Center], Cell[TextData[StyleBox["where", FontSize->14]], "Text", CellMargins->{{16, 19}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{\(a\_n\), "=", RowBox[{ UnderscriptBox["\[Integral]", StyleBox["G", FontSize->11.9375, FontSlant->"Italic"]], \(\(\(f(\[Xi])\)\/\[Xi]\^\(n + 1\)\)\[DifferentialD]\[Xi]\)}]}], FontSize->14], TraditionalForm]], "Input", CellMargins->{{Inherited, 19}, {Inherited, Inherited}}, TextAlignment->Center], Cell[TextData[{ StyleBox[ "is neccesary and sufficient for the analytic continuation of the function \ ", FontSize->14], StyleBox["f", FontSize->14, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" from the chord ", FontSize->14], StyleBox["G", FontSize->14, FontSlant->"Italic"], StyleBox[" into the domain ", FontSize->14], StyleBox["D", FontSize->14, FontSlant->"Italic"], StyleBox[ ". 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Consider \ the strip ", StyleBox["K", FontSlant->"Italic"], " = { ", StyleBox["z", FontSlant->"Italic"], " : 0 < ", StyleBox["\[GothicCapitalR]\[GothicE](z", 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"], ". 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It is important to mention that all calculations were done on a \ regular Pentium machine. We are sure that was possible just because our \ novel and inventive theoretical approach to the problem is much simpler and \ more suitable for numerical testing of the Riemann Hypothesis.", FontSlant->"Plain"]], "Text", CellMargins->{{14, 19}, {Inherited, Inherited}}, FontSize->14, FontSlant->"Italic"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "1. L.A. Aizenberg and A.M. Kutmanov, ", StyleBox[ "On the Possibility of Holomorphic Extension into a Domain of Functions \ Defined on a Connected Piece of its Boundary", FontSlant->"Italic"], ", Math. USSR Sbornik, 2, ", StyleBox["72", FontWeight->"Bold"], " (1992), 467-483." }], "Reference", CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->14], Cell[TextData[{ "2. L. Aizenberg, ", StyleBox["Carleman's Formulas in Complex Analysis,", FontSlant->"Italic"], " Kluwer Acad. Publ., Dozdrecht, 1993." }], "Reference", CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->14], Cell[TextData[{ "3. L. Aizenberg, ", StyleBox["Carleman's Formulas and Conditions of Analytic Extendability", FontSlant->"Italic"], ", Banach Center Publ., ", StyleBox["31", FontWeight->"Bold"], "(1995), 37-34." }], "Reference", CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->14], Cell[TextData[{ "4. J. van de Lune, H.J.J. te Riele, and D.T. Winter, ", StyleBox[ "On the Zeroes of the Riemann Zeta Function in the Critical Strip", FontSlant->"Italic"], ", IV, Math. Comp., ", StyleBox["46", FontWeight->"Bold"], " (1986), 667-681." }], "Reference", FontSize->14], Cell[TextData[{ "5. A.M. Odlyzko, ", StyleBox["The ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`10\^20\)]], StyleBox[" zero of the Riemann Zeta Function and its Neighbors", FontSlant->"Italic"], ", preprint, 1989." }], "Reference", CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->14], Cell[TextData[{ "6. R. Varga, ", StyleBox["Theoretical and Computational Aspects of the Riemann Hypothesis", FontSlant->"Italic"], ", chapter 3 in the book: Scientific Computations on Mathematical Problems \ and Conjectures, Society for Industrial and Applied Mathematics, Philadelhia, \ Pensylvania, 1990." }], "Reference", FontSize->14] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 742}}, WindowToolbars->{"RulerBar", "EditBar"}, Evaluator->"Local", WindowSize->{678, 607}, WindowMargins->{{90, Automatic}, {Automatic, 3}}, Magnification->1, StyleDefinitions -> "ArticleClassic.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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