In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...
, the prime omega functions
and
count the number of prime factors of a natural number
Thereby
(little omega) counts each ''distinct'' prime factor, whereas the related function
(big omega) counts the ''total'' number of prime factors of
honoring their multiplicity (see
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
). That is, if we have a
prime factorization of
of the form
for distinct primes
(
), then the respective prime omega functions are given by
and
. These prime factor counting functions have many important number theoretic relations.
Properties and relations
The function
is
additive
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with f ...
and
is
completely additive.
If
divides
at least once we count it only once, e.g.
.
If
divides
times then we count the exponents, e.g.
. As usual,
means
is the exact power of
dividing
.
If
then
is
squarefree
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...
and related to the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
by
:
If
then
is a prime number.
It is known that the average order of the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
satisfies
.
Like many
arithmetic functions
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their de ...
there is no explicit formula for
or
but there are approximations.
An asymptotic series for the average order of
is given by
:
where
is the
Mertens constant and
are the
Stieltjes constants
In mathematics, the Stieltjes constants are the numbers \gamma_k that occur in the Laurent series expansion of the Riemann zeta function:
:\zeta(s)=\frac+\sum_^\infty \frac \gamma_n (s-1)^n.
The constant \gamma_0 = \gamma = 0.577\dots is known a ...
.
The function
is related to divisor sums over the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
and the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
including the next sums.
:
:
:
:
:
:
:
The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of the
primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
can be expressed by a
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
with the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
:
:
A partition-related exact identity for
is given by
:
where
is the
partition function,
is the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
, and the triangular sequence
is expanded by
:
in terms of the infinite
q-Pochhammer symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer symb ...
and the restricted partition functions
which respectively denote the number of
's in all partitions of
into an ''odd'' (''even'') number of distinct parts.
Continuation to the complex plane
A continuation of
has been found, though it is not analytic everywhere. Note that the normalized
function
is used.
:
Average order and summatory functions
An
average order of both
and
is
. When
is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
a lower bound on the value of the function is
. Similarly, if
is
primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...
then the function is as large as
on average order. When
is a
power of 2
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
, then
.
Asymptotics for the summatory functions over
,
, and
are respectively computed in Hardy and Wright as
:
where
is the
Mertens constant and the constant
is defined by
:
Other sums relating the two variants of the prime omega functions include
:
and
:
Example I: A modified summatory function
In this example we suggest a variant of the summatory functions
estimated in the above results for sufficiently large
. We then prove an asymptotic formula for the growth of this modified summatory function derived from the asymptotic estimate of
provided in the formulas in the main subsection of this article above.
To be completely precise, let the odd-indexed summatory function be defined as
:
where