Prime distributions

Number theory is a great source of multisets, perhaps a better source then set systems as not all finite multisets are used there. Numerous number theoretic functions correspond to multiset operations applied to the prime factorization multiset.

• $\omega(n)$ the order of the prime factorization multiset
• $\Omega(n)$ the cardinality of the prime factorization multiset
• d(n) the number of parts of the multiset
• exponent(n) the exponent of the prime factorization multiset
• prime-signature(n) the signature of the prime factorization multiset

Given any set we can produce a multiset from that set by applying a function to each of its members. If this function is irreversible, this produces a multiset whose partition is more coarse then the set that proceeded it, this is why I call this functional coarsification. This can then be transformed to a relative frequency distribution. This allows us to get a relative frequency distribution for the application of any irreversible function to a set, which is an abundant source of relative frequency distributions. For example, we can apply this to the $\omega$ function for the range from one to a thousand.

{0 0.002, 
 1 0.193, 
 2 0.507, 
 3 0.275, 
 4 0.023}

This can then be compared to the application of $\omega$ from one to a hundred thousand. One see that here the distribution tends towards three to a greater extent.

{0 2.0E-5, 
 1 0.097, 
 2 0.33758, 
 3 0.38844, 
 4 0.15855, 
 5 0.01816, 
 6 2.5E-4}

It should be readily apparent that larger numbers tend to have more prime factors. This is because the divisor ordering is a suborder of the total ordering on the natural numbers. This describes the distribution of the $\omega$ function over small ranges but the Erdos-Kac theorem of probabilistic number theory says that omega(n) is normally distributed with variance and mean $log(log(n))$. This is sometimes expressed as a Poisson distribution rather then a normal distribution because the mean and variance are both the same. This describes the rate that the number of prime factors grows at on average. Probabilistic number theory is the perfect mix of multisets and probabilities, and it demonstrates how the two are intertwined as expressed through the positive integers themselves.

In a complete set of multisets (one that is submultiset closed and finite union closed) different atoms appear as independent random variables, this means that in the set of positive integers the primes act like independent random variables. This is because the positive integers are in some sense complete, as all prime factorizations appear. If they didn't this fact from probabilistic number theory certainly wouldn't work.

Noticing that positive integers themselves are multisets we can produce relative frequency distributions from them which describe the extent to which each prime is distributed within the number. For example, given the number twelve we can produce the distribution {2 2/3, 3 1/3} which of course would be the same for 144. This is associated with the values {1/3 2/3} which describe the relative frequencies of the distribution. These frequencies add up to one, so ontologically this is called a unit partition. To explore probabilistic number theory more I described the different unit partition types of different numbers within a range and how these in turn are distributed. Here are the unit fraction partitions of the first one hundred numbers and how they are themselves distributed within that range.

{*{1/3 1/3 1/3} 0.05,
 *{2/5 3/5} 0.01, 
 *{} 0.01, 
 *{1/5 4/5} 0.02, 
 *{1/2 1/2} 0.32, 
 *{1/4 1/4 1/2} 0.03, 
 *{1/3 2/3} 0.15, 
 *{1/4 3/4} 0.05, 
 *{1/6 5/6} 0.01, 
 *{1} 0.35}

A regular number, that is to say a number with a regular multiset for its prime factorization, is also a square-free power. As we can see here the main unit fraction types described at this range are the regular distributions of order one or two. The unit fraction distribution produces a finer partition then the omega function, so the different partitions are distributed in a way corresponding to the number of distinct prime factors. An antiregular number will have a unit partition that is a set rather then simply a multiset. Rather the number is a squarefree power, antiregular, or of a certain order is determined by the prime distribution of the number. Here are some more examples of prime distributions.

• The prime distribution of 360 is {3 1/3, 2 1/2, 5 1/6} this is antiregular.
• The prime distribution of 384 is {3 1/8, 2 7/8} which is highly irregular, as one prime is far more frequent then the other.
• The prime distribution of 900 is {2 1/3, 3 1/3, 5 1/3} which is regular because this is a squarefree power.
• The prime distribution of 1260 is {7 1/6, 3 1/3, 2 1/3, 5 1/6}.

We talked about the unit partitions and the $\omega$ function dealing with the distinct primes, but the different prime distributions themselves are distributed within the natural numbers in a certain fashion. This isn't too hard to find out, since each prime distribution can be expressed as an exponential equation $b^n$ for some base $b$ which has coprime multiplicities of prime factors, or in other words which is of exponent one. The counting function is therefore logarithmic, and the probability can be computed relatively from that. \[ \frac{log(n)}{log(b)*n} \] This grows asymptotically at degree negative one, which means that individual prime distributions rarely occur (as can also be inferred by their logarithmic counting function). There are an infinite number of prime distributions distributed within the positive integers, but each of them occurring relatively rarely but infinitely nonetheless.

Notice that the natural numbers can be represented as multisets of a single element, which means that the total ordering of natural numbers is simply the inclusion relation on them, then the divisibility relation is simply the relation of even divisibility of these multisets. Next, we can introduce the concept of even disibility of the prime factorization multiset instead. Another way of expressing this is that there exists a positive integer $n$ such that $a^n = b$. In this partial order, the connected components are the numbers with the same prime distributions, as only numbers with the same prime distributions can be exponentiated to get one another. This only occurs when the exponent of two numbers with the same prime distribution divide one another. This generalizes divisibilty ordering associated with multiplication to an ordering associated with exponentation. Each of these partial orders is a suborder of one another.