Function that can be written as a sum over prime factors
In
number theory, an additive function is an
arithmetic function f(n) of the positive
integer variable n such that whenever a and b are
coprime, the function applied to the product ab is the sum of the values of the function applied to a and b:
[1]
Completely additive
An additive function f(n) is said to be completely additive if holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with
totally multiplicative functions. If f is a completely additive function then f(1) = 0.
Every completely additive function is additive, but not vice versa.
Examples
Examples of arithmetic functions which are completely additive are:
- The restriction of the
logarithmic function to
- The multiplicity of a
prime factor p in n, that is the largest exponent m for which pm
divides n.
- a0(n) – the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n (sequence
A001414 in the
OEIS). For example:
- a0(4) = 2 + 2 = 4
- a0(20) = a0(22 · 5) = 2 + 2 + 5 = 9
- a0(27) = 3 + 3 + 3 = 9
- a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14
- a0(2000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
- a0(2003) = 2003
- a0(54,032,858,972,279) = 1240658
- a0(54,032,858,972,302) = 1780417
- a0(20,802,650,704,327,415) = 1240681
- The function Ω(n), defined as the total number of
prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" (sequence
A001222 in the
OEIS). For example;
- Ω(1) = 0, since 1 has no prime factors
- Ω(4) = 2
- Ω(16) = Ω(2·2·2·2) = 4
- Ω(20) = Ω(2·2·5) = 3
- Ω(27) = Ω(3·3·3) = 3
- Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6
- Ω(2000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7
- Ω(2001) = 3
- Ω(2002) = 4
- Ω(2003) = 1
- Ω(54,032,858,972,279) = Ω(11 ⋅ 19932 ⋅ 1236661) = 4 ;
- Ω(54,032,858,972,302) = Ω(2 ⋅ 72 ⋅ 149 ⋅ 2081 ⋅ 1778171) = 6
- Ω(20,802,650,704,327,415) = Ω(5 ⋅ 7 ⋅ 112 ⋅ 19932 ⋅ 1236661) = 7.
Examples of arithmetic functions which are additive but not completely additive are:
- ω(4) = 1
- ω(16) = ω(24) = 1
- ω(20) = ω(22 · 5) = 2
- ω(27) = ω(33) = 1
- ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2
- ω(2000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2
- ω(2001) = 3
- ω(2002) = 4
- ω(2003) = 1
- ω(54,032,858,972,279) = 3
- ω(54,032,858,972,302) = 5
- ω(20,802,650,704,327,415) = 5
- a1(n) – the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence
A008472 in the
OEIS). For example:
- a1(1) = 0
- a1(4) = 2
- a1(20) = 2 + 5 = 7
- a1(27) = 3
- a1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5
- a1(2000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7
- a1(2001) = 55
- a1(2002) = 33
- a1(2003) = 2003
- a1(54,032,858,972,279) = 1238665
- a1(54,032,858,972,302) = 1780410
- a1(20,802,650,704,327,415) = 1238677
Multiplicative functions
From any additive function it is possible to create a related
multiplicative function which is a function with the property that whenever and are coprime then:
One such example is
Summatory functions
Given an additive function , let its summatory function be defined by . The average of is given exactly as
The summatory functions over can be expanded as where
The average of the function is also expressed by these functions as
There is always an absolute constant such that for all
natural numbers ,
Let
Suppose that is an additive function with
such that as ,
Then where is the
Gaussian distribution function
Examples of this result related to the
prime omega function and the numbers of prime divisors of shifted primes include the following for fixed where the relations hold for :
See also
References
-
^ Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207.
online
Further reading
- Janko Bračič, Kolobar aritmetičnih funkcij (
Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97–108) (MSC (2000) 11A25)
- Iwaniec and Kowalski, Analytic number theory, AMS (2004).