16 1. Arithmetic Functions

1.5. Sums over divisors

We remind the reader that an arithmetic function f is a mapping f : N → C.

Some arithmetic functions such as log arise by restricting functions of a

real variable to the positive integers. But in most cases of interest f(n) is

determined by arithmetical information about the integer n. The divisor

function d(n) given as the number of positive divisors of the positive integer

n is an example. Each positive integer n has a unique factorization

n = p1

α1

· · ·pr

αr

into primes and the divisors d of n are the integers of the form

d = p11

β

· · ·pr

βr

where the βj are integers satisfying 0 ≤ βj ≤ αj for j = 1, 2,...,r. Hence

d(n) = (α1 +1)(α2 +1) · · · (αr +1) is a formula for the divisor function d(n)

given in terms of the prime factorization of n.

An arithmetic function f is additive if f(mn) = f(m) + f(n) when-

ever gcd(m, n) = 1. It is multiplicative if f ≡ 0 and f(mn) = f(m)f(n)

whenever gcd(m, n) = 1. It is totally additive or totally multiplicative if the

corresponding property holds without requiring the condition gcd(m, n) = 1.

A multiplicative or additive function can be unambiguously prescribed by

giving its values on the prime powers, and a totally multiplicative or totally

additive function by giving its values on the primes. Note also that f(1) = 1

if f is multiplicative.

The function log(n) and the function Ω(n) that counts the prime divisors

of n with multiplicity are totally additive. The function ω(n) that counts

the distinct prime divisors of n is additive, but not totally additive. The

identity function id given by n → n is totally multiplicative. So is the

Liouville function

λ(n) =

(−1)Ω(n).

The divisor function d(n) is multiplicative, but not totally multiplicative.

The Euler phi-function φ(n) is also multiplicative. Another multiplicative

arithmetic function is the radical

rad(n) =

p|n

p.

It is also called the squarefree kernel.

The von Mangoldt function Λ(n) is an important arithmetic function

that is neither additive nor multiplicative.

The product of two multiplicative functions is multiplicative and the

sum of two additive functions is additive. But a more important algebraic