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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''<sub>''q''</sub>[''X'']}}==
Let {{math|1=''A'' = ''F''<sub>''q''</sub>[''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
| |||