Editing Multiplicative function (section)

== Busche-Ramanujan identities ==

A multiplicative function <math>f</math> is said to be specially multiplicative
if there is a completely multiplicative function <math>f_A</math> such that
:<math>
f(m) f(n) = \sum_{d\mid (m,n)} f(mn/d^2) f_A(d)
</math>
for all positive integers <math>m</math> and <math>n</math>, or equivalently 
:<math>
f(mn) = \sum_{d\mid (m,n)} f(m/d) f(n/d) \mu(d) f_A(d)
</math>
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, 
</math>
and, in 1915, S. Ramanujan gave the inverse form
:<math>
\sigma_k(mn) = \sum_{d\mid (m,n)} \sigma_k(m/d) \sigma_k(n/d) \mu(d) d^k
</math>
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).