Multiplicative function: Difference between revisions

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This property of multiplicative functions significantly reduces the need for computation, as in the following examples for ''n'' = 144 = 2⁴ · 3²:

<math display="block">d(144) = \sigma_0(144) = \sigma_0(2^4) \, \sigma_0(3^2) = (1^0 + 2^0 + 4^0 + 8^0 + 16^0)(1^0 + 3^0 + 9^0) = 5 \cdot 3 = 15</math>

<math display="block">\sigma(144) = \sigma_1(144) = \sigma_1(2^4) \, \sigma_1(3^2) = (1^1 + 2^1 + 4^1 + 8^1 + 16^1)(1^1 + 3^1 + 9^1) = 31 \cdot 13 = 403</math>

<math display="block">\sigma^*(144) = \sigma^*(2^4) \, \sigma^*(3^2) = (1^1 + 16^1)(1^1 + 9^1) = 17 \cdot 10 = 170</math>

Similarly, we have:

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f=g_1\ast\cdots\ast g_r\ast h_1^{-1}\ast\cdots\ast h_s^{-1},

</math>

where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order <math>(1, 1)</math> are known as totient functions, and rational arithmetical functions of order <math>(2,0)</math> are known as quadratic functions or specially multiplicative functions. Euler's function <math>\varphi(n)</math> is a totient function, and the divisor function <math>\sigma_k(n)</math> is a quadratic function.

Completely multiplicative functions are rational arithmetical functions of order <math>(1,0)</math>. Liouville's function <math>\lambda(n)</math> is completely multiplicative. The Möbius function <math>\mu(n)</math> is a rational arithmetical function of order <math>(0, 1)</math>.

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

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for all positive integers <math>m</math> and <math>n</math>, where <math>\mu</math> is the Möbius function.

These are known as Busche-Ramanujan identities.

In 1906, E. Busche stated the identity

:<math>

\sigma_k(m) \sigma_k(n) = \sum_{d\mid (m,n)} \sigma_k(mn/d^2) d^k,

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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>. In fact, quadraticQuadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).

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

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== Generalizations ==

An arithmetical function <math>f</math> is

quasimultiplicative if there exists a nonzero constant <math>c</math> such that

<math>

c\,f(mn)=f(m)f(n)

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(under the convention that <math>f_m(x)=0</math> if <math>x</math> is not a positive integer.) This concept is due to David Rearick (1966).

An arithmetical function <math>f</math> is Selberg multiplicative if

for each prime <math>p</math> there exists a function <math>f_p</math> on nonnegative integers with <math>f_p(0)=1</math> for

all but finitely many primes <math>p</math> such that

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*{{cite journal

|author=P. Haukkanen

|title=Some characterizations of specially multiplicative functions

|journal=Int. J. Math. Math. Sci.

|volume=2003

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|issue=37

|doi=10.1155/S0161171203301139

|doi-access=free

|url=https://www.emis.de/journals/HOA/IJMMS/Volume2003_37/515979.abs.html

}}

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|issue=3

|pages=316–317

|year=1972 |doi=10.1007/BF01844515

}}

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|volume=33

|pages=49–53

|year=1966 }}

*{{cite journal

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|volume=9

|pages=1301–1311

|year=2013 |issue=5

|doi=10.1142/S1793042113500280

|arxiv=1301.3331