Content deleted Content added
No edit summary |
|||
Line 164:
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 ==
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)
</math>
for all positive integers <math>m, n</math> with <math>(m, n)=1</math>.
It is easy to see that an arithmetical function <math>f</math> not identically zero is quasimultiplicative if and only if <math>f(1)\ne 0</math> and
<math>
f(1)f(mn)=f(m)f(n)
</math>
for all <math>m, n</math> with <math>(m, n)=1</math>. Then <math>c=f(1)</math>.
Quasimultiplicative functions are multiplicative functions
multiplied by a constant.
An arithmetical function <math>f</math> not identically zero is quasimultiplicative if and only if <math>f(1)\ne 0</math> and <math>f/f(1)</math> is multiplicative.
D. B. Lahiri introduced the concept of quasimultiplicative functions as a special case of hypomultiplicative functions.
The concept of a multiplicative function or a quasimultiplicative function is not satisfactory in the sense that compositions such as <math>f(kn), f(k/n), f(n/k), f([k, n])</math> <math>(k\ne 1)</math> preserve neither multiplicativity nor quasimultiplicativity. This has led to the concepts of semimultiplicative and Selberg multiplicative functions.
An arithmetical function <math>f</math> is said to be semimultiplicative
if there exists a nonzero constant <math>c</math>, a positive integer <math>a</math> and
a multiplicative function <math>f_m</math> such that
<math>
f(n)=c f_m(n/a)
</math>
for all <math>n</math>.
(Here <math>f_m(x)=0</math> if <math>x</math> is not a positive integer.)
This concept is due to David Rearick.
An arithmetical function <math>f</math> not identically zero is semimultiplicative if and only if
there exists a positive integer <math>a</math> such that
<math>f(a)\ne 0</math> and
<math>f(a x)</math> is an arithmetical function (i.e. <math>f(a x)=0</math> if <math>x</math> is not a positive integer) and
<math>
\frac{f(a n)}{f(a)}
</math>
is multiplicative in <math>n</math>.
A nice characterization is as follows.
An arithmetical function <math>f</math> is semimultiplicative if and only if
<math>
f(m)f(n)=f((m, n))f([m, n])
</math>
for all positive integers <math>m, n</math>.
Semimultiplicative functions <math>f</math> with <math>a=1</math> and <math>c=1</math> (i.e. <math>f(1)=1</math>) are multiplicative functions and
semimultiplicative functions <math>f</math> with <math>a=1</math> (i.e. <math>f(1)\ne 0</math>) are quasimultiplicative functions.
Atle Selberg said on the concept of the usual multiplicative functions that ``I have never been very satisfied with this definition and would prefer to define a multiplicative function as follows".
This leads to the following concept.
An arithmetical function <math>f</math> is Selberg multiplicative if
for each prime <math>p</math> there exists <math>F_p</math>
with <math>F_p(0)=1</math> for
all but finitely many primes <math>p</math> such that
<math>
f(n)=\prod_{p} F_p(\nu_p(n))
</math>
for all <math>n</math>, where <math>\nu_p(n)</math> is the exponent of <math>p</math> in the canonical factorization of <math>n</math>.
Multiplicative functions <math>f</math> are Selberg multiplicative with
<math>
F_p(\nu_p(n))=f(p^{\nu_p(n)}).
</math>
Quasimultiplicative functions <math>f</math> are Selberg multiplicative with
Selberg factorization
<math>
f(n)=f(1)\prod_{p}
\left(\frac{f(p^{\nu_p(n)})}{f(1)}\right)
</math>
provided that <math>f(1)\ne 0</math>.
Finally, an arithmetical function is Selberg multiplicative if and only
if it is semimultiplicative. In fact, a semimultiplicative function <math>f</math> possesses a Selberg factorization as
<math>
f(n)=f(a)\prod_{p}
\left(\frac{f(ap^{\nu_p(n)-\nu_p(a)})}{f(a)}\right).
</math>
==See also==
Line 173 ⟶ 255:
* See chapter 2 of {{Apostol IANT}}
* {{cite journal |title=Efficient estimation of a multivariate multiplicative volatility model |journal=Journal of Econometrics |date=2010 |volume=159 |issue=1 |pages=55–73 |doi=10.1016/j.jeconom.2010.04.007 |s2cid=54812323 |url=http://sticerd.lse.ac.uk/dps/em/em541.pdf |last1=Hafner |first1=Christian M. |last2=Linton |first2=Oliver }}
*{{cite journal
|author=R. Vaidyanathaswamy
|title=The theory of multiplicative arithmetic functions
|journal=Transactions of the American Mathematical Society
|volume=33
|issue=2
|pages=579-662
|year=1931
|doi=10.1090/S0002-9947-1931-1501607-1
|doi-access=free }}
==External links==
| |||