Arithmetic Functions

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Arithmetic Function

ARITHMETIC FUNCTION

A function \(f\) defined on \(\mathbb{N}\) is called arithmetic.

\(f : \mathbb{N} \to \mathbb{C}\)

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Additive Function

ADDITIVE FUNCTION

An arithmetic function \(f\) is called additive if

\(f(mn) = f(m) + f(n)\)

for all \(m\) and \(n\) with \(\gcd(m, n) = 1\).

Being completely additive means that the coprime condition can be dispensed with.

The prime omega function \(\omega(n)\) is additive and the prime omega function \(\Omega(n)\) is completely additive.

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Multiplicative Function

MULTIPLICATIVE FUNCTION [Vaughan Definition 3.13]

If an arithmetical function \(f\) satisfies

\(f(mn) = f(m) f(n)\)

whenever \(\gcd(m, n) = 1\) then \(f\) is called multiplicative.

Being completely multiplicative means that the coprime condition can be dispensed with.

The divisor function \(d(n)\) is multiplicative and \(\sqrt{n}\) is completely multiplicative.

Besides the zero function a multiplicative function \(f\) satisfies \(f(1) = 1\) and \(f(n)\) is then determined by the values it takes at the prime powers

\(f(n) = f(p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}) = f(p_1^{a_1}) f(p_2^{a_2}) \dots f(p_k^{a_k})\) with \(p_i \in \mathbb{P}\) and \(a_i \in \mathbb{N}\)

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Prime Omega Functions

The prime omega functions \(\omega(n)\) and \(\Omega(n)\) are additive and completely additive arithmetic functions, respectively, which count the number of prime factors of a natural number.

PRIME OMEGA FUNCTIONS

Suppose we have a prime factorization of \(n\) of the form

\(n = p_1^{\alpha_1} p_2^{\alpha_2} \dots p_k^{\alpha_k}\)

for distinct primes \(p_i\) with \(1 \le i \le k\).

Then \(\omega(n)\) counts the number of distinct prime factors.

\( \begin{aligned} \omega(n) = k = \sum_{p \mid n} 1 \\ \end {aligned} \)

And \(\Omega(n)\) counts the total number of prime factors, honoring their multiplicity.

\( \begin{aligned} \Omega(n) = \sum_{i=1}^k \alpha_i = \sum_{p^\alpha \mid n} 1 = \sum_{p^\alpha \| n} \alpha \\ \end {aligned} \)

\(\omega(n) \le \Omega(n)\)

If \(\omega(n) = 1\) then \(n\) is a prime power and if \(\Omega(n) = 1\) then \(n\) is a prime.

EXAMPLE

\( \begin{aligned} n & && \omega(n) && \Omega(n) \\ \hline 2 &= 2^1 && 1 && 1 \\ 3 &= 3^1 && 1 && 1 \\ 4 &= 2^2 && 1 && 2 \\ 5 &= 5^1 && 1 && 1 \\ 6 &= 2^1 \times 3^1 && 2 && 2 \\ 7 &= 7^1 && 1 && 1 \\ 8 &= 2^3 && 1 && 3 \\ 9 &= 3^2 && 1 && 2 \\ 10 &= 2^1 \times 5^1 && 2 && 2 \\ 11 &= 11^1 && 1 && 1 \\ 12 &= 2^2 \times 3^1 && 2 && 3 \\ 13 &= 13^1 && 1 && 1 \\ 14 &= 7^2 && 1 && 2 \\ 15 &= 3^1 \times 5^1 && 2 && 2 \\ 16 &= 2^4 && 1 && 4 \\ 17 &= 17^1 && 1 && 1 \\ 18 &= 2^1 \times 3^2 && 2 && 3 \\ 19 &= 19^1 && 1 && 1 \\ 20 &= 2^2 \times 5^1 && 2 && 3 \\ 21 &= 3^1 \times 7^1 && 2 && 2 \\ 22 &= 2^1 \times 11^1 && 2 && 2 \\ 23 &= 23^1 && 1 && 1 \\ 24 &= 2^3 \times 3^1 && 2 && 4 \\ 25 &= 5^2 && 1 && 2 \\ 26 &= 2^1 \times 13^1 && 2 && 2 \\ 27 &= 3^3 && 1 && 3 \\ 28 &= 2^2 \times 7^1 && 2 && 3 \\ 29 &= 29^1 && 1 && 1 \\ 30 &= 2^1 \times 3^1 \times 5^1 && 3 && 3 \\ \end {aligned} \)

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Sum Function

SUM FUNCTION

Let

\( \begin{aligned} \sum_{d \mid n} \end {aligned} \)

denote summation over the divisors of \(n\) so that for a given arithmetic function \(f(n)\) we have a new function called the sum function \(F(n)\) defined by

\( \begin{aligned} F(n) = \sum_{d \mid n} f(d) \end {aligned} \)

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