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>

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By convention, the identity element <math>\varepsilon</math> under the Dirichlet convolution is a rational arithmetical function of order <math>(0, 0)</math>.

All rational arithmetical functions are multiplicative. A multiplicative function ''f'' is a rational arithmetical function of order <math>(r, s)</math> [[if and only if]] its Bell series is of the form

<math display="block">

{\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}=

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for <math>k=0</math>. S. Chowla gave the inverse form for general <math>k</math> in 1929, see P. J. McCarthy (1986). The study of Busche-Ramanujan identities begun from an attempt to better understand the special cases given by Busche and Ramanujan.

It is known that quadratic functions <math>f=g_1\ast g_2</math> satisfy the Busche-Ramanujan identities with <math>f_A=g_1g_2</math>. Quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see [[Ramaswamy S. Vaidyanathaswamy|R. Vaidyanathaswamy]] (1931).

==Multiplicative function over {{math|''F''_''q''[''X'']}}==

Let {{math|1=''A'' = ''F''_''q''[''X'']}}, the [[polynomial ring]] over the [[finite field]] with ''q'' elements. ''A'' is a [[principal ideal ___domain]] and therefore ''A'' is a [[unique factorization ___domain]].

A complex-valued function <math>\lambda</math> on ''A'' is called '''multiplicative''' if <math>\lambda(fg)=\lambda(f)\lambda(g)</math> whenever ''f'' and ''g'' are [[relatively prime]].

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Thus it gives an estimate value of <math display="block">L_t(\tau;u) = \sum_{t=1}^T K_h(u - t/T)\begin{bmatrix} ln\tau + \frac{y^2_t}{g_t\tau} \end{bmatrix}</math>

with a local [[likelihood function]] for <math>y^2_t</math> with known <math>g_t</math> and unknown <math>\sigma^2(t/T)</math>.

== Generalizations ==

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*{{cite journal

|author=R. Vaidyanathaswamy

|author-link=Ramaswamy S. Vaidyanathaswamy

|title=The theory of multiplicative arithmetic functions

|journal=Transactions of the American Mathematical Society