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Concept
Prime omega function
Résumé
In number theory, 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). 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.
The function is additive 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 and related to the Möbius function by
If then is a prime number.
It is known that the average order of the divisor function satisfies .
Like many arithmetic functions 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.
The function is related to divisor sums over the Möbius function and the divisor function including the next sums.
The characteristic function of the primes can be expressed by a convolution with the
Möbius function:
A partition-related exact identity for is given by
where is the partition function, is the Möbius function, and the triangular sequence is expanded by
in terms of the infinite q-Pochhammer symbol and the restricted partition functions which respectively denote the number of 's in all partitions of into an odd (even) number of distinct parts.
A continuation of has been found, though it is not analytic everywhere. Note that the normalized function is used.
An average order of both and is . When is prime a lower bound on the value of the function is . Similarly, if is primorial then the function is as large as
on average order.
Source officielle
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