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CHAPTER 7 - Multiplicative Functions
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7.1 The Euler's Phi-Function

 

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An applet for computing values of the Euler phi function can be found at http://www.math-it.de/Mathematik/Zahlentheorie/Zahl/ZahlApplet.html

 

You can use the tool at http://www.numbertheory.org/php/factor.html to compute values of the Euler phi function.

 

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Biographical information about Joseph Liouville can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Liouville.html (Liouville)

 

 

7.2 The Sum and Number of Divisors

 

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You can find the sum and number of divisors of a positive integer, as well as compute the values of several other multiplicative functions, using the tool at http://www.numbertheory.org/php/factor.html

 

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Biographical information about Srinivasa Ramanujan can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Ramanujan.html (Ramanujan)

 

 

7.3 Perfect Numbers and Mersenne Primes

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A survey article about perfect numbers can be found at the St. Andrews History of Mathematics site at
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html (Perfect Numbers)

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A wealth of information about Mersenne primes can be found at
http://www.utm.edu/research/primes/mersenne.shtml (Mersenne Primes: History, Theorems, and Lists)
Biographical information about Marin Mersenne can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Mersenne.html (Mersenne)

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Biographical information about François-Edouard-Anatole Lucas can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Lucas.html (Lucas)

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The history of the search for Mersenne primes is described in detail at
http://www.utm.edu/research/primes/mersenne.shtml (Mersenne Primes: History, Theorems and Lists)
Luke Welsh, one of the discoverers of the 29th Mersenne prime, has an excellent site containing a wealth of information about Marin Mersenne and the search for Mersenne primes at
http://www.scruznet.com/~luke/mersenne.htm (Luke's Marin Mersenne Page)

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You can learn about the Great Internet Mersenne Prime Search, download software to look for new Mersenne primes, and join the search itself at
http://www.mersenne.org (Mersenne Prime Search)
You can learn about progress with the search for Mersenne primes over PrimeNet and obtain software for joining PrimeNet which is associated with the Great Internet Prime Search at
http://www.entropia.com/ips/ (Internet PrimeNet Server)

 

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Conditions that odd perfect numbers must satisfy are described at http://mathworld.wolfram.com/OddPerfectNumber.html (Eric W. Weisstein. "Odd Perfect Number." From MathWorld--A Wolfram Web Resource)

 

To learn more about the prizes offered by the Electronic Frontier Foundation for finding large primes go to http://www.eff.org/awards/coop.html (EFF Cooperative Computing Awards)

 

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An applet that determines whether a positive integer is abundant, perfect, or deficient can be found at http://ccins.camosun.bc.ca/~jbritton/perfect/jbperfect.htm (Perfect Number Analyzer)

To learn more about amicable numbers you can consult
http://xraysgi.ims.uconn.edu/amicable.html (Perfect, amicable, and social numbers)

To access a list of all known amicable pairs go to http://amicable.homepage.dk/knwnap.htm  (Known Amicable Pairs)  

Information about multiply perfect numbers can be found at
http://www.uni-bielefeld.de/~achim/mpn.html (Multiply Perfect Numbers)

 

7.4 Möbius Inversion

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Biographical information about August Ferdinand Möbius can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Mobius.html (Möbius)

 

 

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You can find the paper by Odlyzko and te Riele disproving the Merten’s conjecture at http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf (Disproof of the Merten’s Conjecture)