Riemann Zeta Function: Important Number Relationships (10)

We earlier looked at how the Riemann zeta function (for positive integer values of s) has a role to play with respect to the maximum frequency that any prime can occur (with respect to the unique factor compositions of the natural numbers).

We concluded then the probability that a number chosen at random will have a factor composition with at least one prime repeating n or more times  = 1 – 1/ζ(n).
Alternatively we can state that the probability that a number chosen at random will not contain a factor composition where at least one prime repeats n or more times = 1/ζ(n).

Therefore, for example, the probability that such a random number will have at least one prime factor repeating 2 or more times = 1 – 1/ζ(2) = 1 – .6079 = .3921 (approx).

Alternatively we could express this by stating that that the probability that such a number will have no prime factor repeating more than once =  1/ζ(2) = .6079 (approx).

We also derived expressions - again based on the Riemann zeta function for the positive integer values - that express the probability that a number chosen at random will contain at most n repeating prime factors.

So again this general probability is given as 1/ζ(n + 1) – 1/ζ(n).

So therefore to illustrate, the probability that a number chosen at random will contain at most 2 repeating prime factors = 1/ζ(3) – 1/ζ(2)  = .8319 - .6079 = .224 (approx).

However these measurements are confined to the external aspect of the number system. So basically in this case, we are trying to ascertain the relative frequency (with respect to the overall system) that certain numbers occur with such stipulated prime factor compositions.

Therefore we can state - using our most recent example - that 22.4% of all numbers will contain factor combinations, where at most one prime repeats twice!

But as I was anxious to emphasise in my previous entries, based on recognition of both the Type 1 and Type 2 aspects of the number system, that a complementary interpretation (entailing the Riemann zeta function for the positive integers) will also necessarily exist in an internal number fashion.

So therefore in this context, we are looking at the frequency with respect to the overall occurrence of factors associated with such stipulated prime combinations.

Now, there is little point in trying to count in absolute terms the total number of factors associated  (as this total will necessarily increase without limit for numbers associated with all factor compositions).

However we can make meaningful comparisons by comparing the average number of factors for each prime combination (at a given appropriately high level of the number system).

We can then for example compare the average number of factors belonging to those numbers which contain - say - at most a prime that occurs just once (i.e. no repeating prime) with the average count of factors belonging to those numbers where one (or more) primes repeat at most 2 times.

Now one would reasonably expect to find, on average, more factors belonging to those numbers where one or more primes can repeat twice compared to those where no prime factor can repeat more than once!

And this ratio is given it would appear (as based on preliminary sample evidence) as ζ(2).

Alternatively in reverse, the average ratio of the total frequency of factors associated with those numbers, where no prime repeats more than once, in relation to the total frequency of factors associated with numbers where one (or more) primes repeat at most twice = 1/ζ(2).

And in like fashion, the average ratio of the total frequency of factors associated with numbers where  one (or more) primes repeat at most 2 times in relation to the total frequency of factors for numbers where one (or more) primes repeat at most 3 times = 1/ ζ(3).

And in general terms, the average ratio of the total frequency of factors associated with numbers where one or more primes repeat at most (n – 1) times in relation to the total frequency of factors for numbers where one (or more) primes repeat at most n times =  1/ ζ(n).

And as ζ(n) → 1 (for large n), this means that the corresponding ratio in relation to the total frequency of factors would likewise → 1. So for example is we were to count up the total frequency of factors in a certain  region (appropriately high up the number scale) for all those numbers, where at most one or more primes repeat - say - 7 times and then count up the corresponding total of factors for the same amount of numbers where at most one or more primes repeat 8 times, there would be little difference with respect to the overall total of factors in each case! 

We can also get equivalent expressions where the stipulation - with relation to the factor composition - is that a prime (or primes) repeats more than n times.

Then from this perspective, the average ratio of the total frequency of factors for numbers where a prime (or primes) repeats 1 time or more (which represents 100% of numbers) in relation to the total frequency of factors where a prime (or primes) repeats 2 times or more = {1/ζ(2)}/{1/ζ(3)} = ζ(3)/ζ2).

And in general terms the average ratio of the total frequency of factors for numbers where a prime (or primes) repeats (n – 1) times or more, in relation to the total frequency of factors where a prime (or primes) repeats n times or more = ζ(n + 1)/ζ(n).