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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.

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Average order of an arithmetic function

In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average". Let be an arithmetic function. We say that an average order of is if as tends to infinity. It is conventional to choose an approximating function that is continuous and monotone. But even so an average order is of course not unique. In cases where the limit exists, it is said that has a mean value (average value) .

Divisor sum identities

The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function with one: These identities include applications to sums of an arithmetic function over just the proper prime divisors of .

Lambert series

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resumed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1: This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series.

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