summatory function of arithmetic function

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


where d runs the positive divisorsMathworldPlanetmathPlanetmath of the integer n.

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

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

dnφ(d)=dnφ(nd)=nfor all n+.

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  1mn  where  gcd(m,n)=d.  Then  gcd(m/d,n/d)=1  and  1m/dn/d.  This defines a bijection between the prime classes modulo n/d and such values of m in {1, 2,,n-1} for which  gcd(m,n)=d.  The number of the latters φ(n/d). Furthermore, the only m with  1mn  and  gcd(m,n)=n is  m:=n,  and  φ(n/n)=φ(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