summatory function of arithmetic function

Definition.  The summatory function $F$ of an arithmetic function $f$ is the Dirichlet convolution of $F$ and the constant function 1, i.e.

$$F(n)=:\sum _{d\mid n}f(d)$$

where $d$ runs the positive divisors of the integer $n$.

It may be proved that the summatory function of a multiplicative function is multiplicative.

Theorem.  The summatory function of the Euler phi function is the identity function:

$$\sum _{d\mid n}\phi (d)=\sum _{d\mid n}\phi \left(\frac{n}{d}\right)=n\mathit{\hspace{1em}}\text{for all }n\in {\mathbb{Z}}_{+}.$$

Proof.  The first equality follows from the fact that any positive divisor of $n$ is got from $n/d$ where $d$ is a divisor of $n$. Further, let  $1\le m\le n$  where  $\gcd (m,n)=d$.  Then  $\gcd (m/d,n/d)=1$  and  $1\le m/d\le n/d$.  This defines a bijection between the prime classes modulo $n/d$ and such values of $m$ in $\{1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },n-1\}$ for which  $\gcd (m,n)=d$.  The number of the latters $\phi (n/d)$. Furthermore, the only $m$ with  $1\le m\le n$  and  $\gcd (m,n)=n$ is  $m:=n$,  and  $\phi (n/n)=\phi (1)$, by definition.  Summing then over all possible values $d$ yields the second equality.

Title	summatory function of arithmetic function
Canonical name	SummatoryFunctionOfArithmeticFunction
Date of creation	2013-03-22 19:31:53
Last modified on	2013-03-22 19:31:53
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Theorem
Classification	msc 11A25
Synonym	summatory function
Related topic	EulerPhifunction
Related topic	PrimeClass