Multiplicative function: Difference between revisions

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** <math>\sigma_1(n)=\sigma(n)</math>, the sum of all the positive divisors of <math>n</math>.

*The<math>\sigma^*_k(n)</math>: the sum of the <math>k</math>-th powers of theall [[unitary divisor]]s is denoted byof <math>\sigma^*_k(n)</math>:

::<math>\sigma_k^*(n) \,=\!\!\sum_{d \,\mid\, n \atop \gcd(d,\,n/d)=1} \!\!\! d^k</math>

* <math>a(n)</math>: the number of non-isomorphic [[abelian groups]] of order <math>n</math>

* ''γ''<math>\gamma(''n'')</math>, defined by ''γ''<math>\gamma(''n'') = (&minus;-1)<sup>''ω''^{\omega(n)}</supmath>, where the [[additive function]] ''ω''<math>\omega(''n'')</math> is the number of distinct primes dividing ''<math>n''.</math>

* ''τ''<math>\tau(''n'')</math>: the [[Ramanujan tau function]].

* All [[Dirichlet character]]s are completely multiplicative functions., Forfor example

** <math>(''n''/''p'')</math>, the [[Legendre symbol]], considered as a function of ''<math>n''</math> where ''<math>p''</math> is a fixed [[prime number]].

An example of a non-multiplicative function is the arithmetic function ''r''<sub>2</submath>r_2(''n'') -</math>, the number of representations of ''<math>n''</math> as a sum of squares of two integers, [[positive number|positive]], [[negative number|negative]], or [[0 (number)|zero]], where in counting the number of ways, reversal of order is allowed. For example:

{{block indent|em=1.2|text=1 = 1² + 0² = (−1)² + 0² = 0² + 1² = 0² + (−1)²}}

and therefore ''r''<sub>2</submath>r_2(1) = 4 ≠\neq 1</math>. This shows that the function is not multiplicative. However, ''r''<sub>2</submath>r_2(''n'')/4</math> is multiplicative.

In the [[On-Line Encyclopedia of Integer Sequences]], sequences of values of a multiplicative function have the keyword "mult".<ref>{{cite web | url=http://oeis.org/search?q=keyword:mult | title=Keyword:mult - OEIS }}</ref>