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An Approach to Prime Numbers , the L.C.M. , and the Zeta Functions

by Frank C. Fung - 1st published in May, 2009.

This paper is dedicated to my father , the late Charles Hok-Ling Fung ( 1909-2002 ) , on the occassion of his Centenial Year .

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Summary & Key Findings

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Summary and Key Findings :

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Other Sections of the paper :

Table Of Content

Part I - Introduction and Preliminaries
Section IIntroduction and a-priori Assumptions
Section IIA Special Note on the Notations in this paper
Part II - Understanding the role of the L.C.M.
Section IIISetting up the L.C.M.-based Function { Lamda-of-(M) }
Section IVSetting up the [ QTT-i ]'s for any given value of [ M ]
Section VThe Value { [ M ] raised to the power [ pi-of-(M) ] }
Section VIActual Statistics for [ M = 31 ]
Part III - Tracking Carl Friedrich Gauss
Section VIIA Fast-Track Approach to identify { prime numbers }
Section VIIIArea Under the Hyperbola
Section IXSingularities and Symmetries in 2-D / 3-D / 4-D Hyperbola's
Part IV - Tracking Leonhard Euler
Section XSetting up 3 more Functions
Section XIThe Euler Product Formula via an inequality
Part V - Tracking Georg Friedrich Bernhard Riemann
Section XIIRiemann's start-off with the Euler Observation
Section XIIIThe Zeta Function extended to the [ critical strip ]
Section XIVRiemann's Prime Counting Function F(x) and its derived function f(x)
Section XVFinite Values at [ s = 1 ] for a finite value of [ M ]
Section XVIUnresolved Issues for the author
Part VI - Summarizing and Concluding Remarks
Section XVIISummarizing the 5 prime-related { Discrete Series }
Section XVIIIConcluding Remarks
Appendix
Appenidx AA Listing of the First 600 Prime Numbers
Appenidx BThe 2-Dimension { 42 x 42 Multiplication Table } for Section VII
Epilogs ( added 2009-5-17 / 6-02 )
Epilog IUsing { L.C.M.-based modulo's } in the Fast Track Approach
Epilog IIThe Separability of Riemann's f(x) Function into its component parts
Epilog IIIThe gap in-between { consecutive prime numbers } to reach Infinity

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original dated 2009-5-08 *** Updated 2009-5-17 / 6-02 / 10-26