Additive function: Difference between revisions

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Revision as of 00:10, 19 September 2002

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=== Examples ===

AnArithmetic arithmetical functionfunctions which is additive but notare completely additive isare:

* A contraction of the [[logarithm|logarithmic function]] on '''N'''.

* A function Ω(''n''), defined for every ''n'' ≥ 2 of total number of primes, which devide given positive integer ''n''. We put also Ω(1) = 0. Some values:

::Ω(4) = 2

::Ω(27) = 3

::Ω(2,000) = 2

::Ω(2,001) = 3

::Ω(2,002) = 4

::Ω(2,003) = 1

::Ω(54,032,858,972,279) = 3

::Ω(54,032,858,972,302) = 6

::Ω(20,802,650,704,327,415) = 7

:: ...

An example of an arithmetic function which is additive but not completely additive is:

: ω(''n'') = ∑_''p''|''n'' 1(''n''),

for every positive integer ''n'', where sum runs over all different [[prime number|primes]] that do not devide ''n'' and 1(''n'') is a constant function, defined by 1(''n'') = 1. The ω function tells us how many different primes devide arbitrary positive integer ''n''. Some values (compare with Ω(''n'')):

::ω(4) = 1

::ω(27) = 1

::ω(2,000) = 2

::ω(2,001) = 3

::ω(2,002) = 4

::ω(2,003) = 1

::ω(54,032,858,972,279) = 3

::ω(54,032,858,972,302) = 5

::ω(20,802,650,704,327,415) = 5

:: ...

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=== References ===

'''Sources:'''

# Janko Bra&#269;i&#269;, ''Kolobar aritmeti&#269;nih funkcij'' (''Ring of arithmetical functions''), (Obzornik mat, fiz. '''49''' (2002) 4, pp 97 - 108) <font color=darkblue> (MSC (2000) 11A25) </font>