Groups of Arithetical Functions

The sum and product of multiplicative functions are also multiplicative. Suppose  
\[f, \; g\]
  are multiplicative functions, then  
\[f(nm)=f(n)f(m), \; g(mn)=g(m)g(n)\]

Also if we restrict the set of functions to satisfy  
  the the set of arithmetical functions is a group under multiplication.
\[f(mn)+g(mn)=f(m)f(n)+g(m)g(n) \neq f(m)g(m)f(m)g(n)=\]

so the set of arithmetical functions does not form a group under addition since the sum of arithmetical functions is not an arithmetical function.

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