Arithmetic function: Difference between revisions

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The most important arithmetic functions are the [[additive function|additive]] and the [[multiplicative function|multiplicative]] ones.

An important operation on arithmetic functions is the [[Dirichlet convolution]].

=== Examples ===

The articles on additive and multiplicative functions contain several important examples.

Examples of a non-multiplicative functions are:

The articles on additive and multiplicative functions contain several important examples of arithmetic functions. Here are some examples that are neither additive nor multiplicative:

* ''c''_''4''(''n'') - the number of ways that ''n'' can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

* ''c''_''4''(''n'') - the number of ways that ''n'' can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

::1 = 1²+0²+0²+0² = 0²+1²+0²+0² = 0²+0²+1²+0² = 0²+0²+0²+1²,

:hence ''c''₄(1)=4 ≠ 1.

* ''P''(''n''), the [[Partition function]] - the number of representations of ''n'' as a sum of positive integers, where we don't distinguish between different orders of the summands. For instance: ''P''(2 · 5) = ''P''(10) = 42 and ''P''(2)''P''(5) = 2 · 7 = 14 ≠ 42.

* π (''n''), the [[Prime number theorem|Prime counting function]] - the number of [[prime number|primes]] less than or equal to a given number ''n''. We have π(1) = 0 ≠ 1, π(2 · 5) =and π(10) = 4 and π(2)the π(5)primes =below 110 ·being 2, 3, =5, 3and ≠ 47).

* ''a''_''0''(''n'') - the sum of primes dividing ''n'', sometimes called sopfr(''n''). We have ''a''_''0''(120) = 0 ≠ 1, ''a''_''0''(2² · 5) = ''a''_''0''(10) = 7 and ''a''_''0''(2) ''a''_''0''(5) =+ 2 ·+ 5 = 10 ≠9. 7([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001414 SIDN A001414]).

* ''a''_''1''(''n'') - the sum of the distinct primes dividing ''n'', sometimes called sopf(''n''). We have ''a''_''1''(1) = 0 ≠ 1, ''a''_''1''(2 · 520) = ''a''_''1''(10) = 7 and ''a''_''1''(2) ''a''_''1''(5) = 2 ·+ 5 = 107. ≠ 7([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472 SIDN A008472]).