# 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.

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

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 $\mathrm{gcd}(m,n)=d$. Then $\mathrm{gcd}(m/d,n/d)=1$ 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 $\mathrm{gcd}(m,n)=d$. The number of the latters $\phi (n/d)$.
Furthermore, the only $m$ with $1\le m\le n$ and $\mathrm{gcd}(m,n)=n$ is $m:=n$, and $\phi (n/n)=\phi (1)$, by definition. Summing then over all possible values $d$ yields the second equality.

## References

- 1 Peter Hackman: Elementary number theory. HHH productions, Linköping (2009).

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 |