## Friday, November 11, 2016

### Riemann Zeta Function: Important Number Relationships (12)

Just to illustrate a little better the terminology, I have been using in recent blog entries with respect to the distribution of the prime factors of the natural numbers, I include below the factor breakdown for the 16 consecutive numbers between 800000000000001 and 800000000000016.

8000000000000001 = 4447 * 1423 * 409 * 163 * 43 * 7 * 7 * 3 * 3
8000000000000002 = 23121387283237 * 173 * 2
8000000000000003 = 159897739 * 112939 * 443
8000000000000004 = 37884167 * 138563 * 127 * 3 * 2 * 2
8000000000000005 = 3470715835141 * 461 * 5
8000000000000006 = 601775236949 * 23 * 17 * 17 * 2
8000000000000007 = 33471823001 * 79669 * 3
8000000000000008 = 9091 * 2161 * 241 * 211 * 13 * 11 * 7 * 2 * 2 * 2
8000000000000009 = 556261 * 42683 * 107 * 67 * 47
8000000000000010 = 88888888888889 * 5 * 3 * 3 * 2
8000000000000011 = 8000000000000011 * 1
8000000000000012 = 105263157894737 * 19 * 2 * 2
8000000000000013 = 99826551367 * 26713 * 3
8000000000000014 = 9364339 * 468883 * 911 * 2
8000000000000015 = 20476333 * 11162713 * 7 * 5
8000000000000016 = 166666666666667 * 3 * 2 * 2 * 2 * 2

Now the factors of the first number belong to that class where 1 or more primes occur at most 2 times. As we can see both 7 and 3 occur here twice (with all other factors occurring just once).

As we have seen for the number system as a whole, the proportion of all numbers belonging to each of the designated factor classes is governed by the Riemann zeta function (for the positive integers).

The proportion belonging to this class i.e. where again 1 or more prime factors occur at most 2 times = 1/ζ(3) – 1/ζ(2) = .224 (approx) i.e. 22.4%. For convenience we will refer to this as Class 2.

The factors of the second number belong to the most frequent class of numbers, where each prime occurs on just 1 occasion (i.e. where no prime occurs more than once).

The proportion belonging to this class = 1/ζ(2) – 1/ζ(1) = .608 (approx) i.e. 60.8%. For convenience we will refer to this as Class 1.

The prime factors of the third number again belongs to to Class 1 (where each prime occurs just once).

The prime factors of the 4th number then belong to Class 2. (On this occasion only 1 prime occurs at most twice!)

The prime factors of the 5th number again belong to Class 1; the factors of the 6th belong to Class 2 and the factors of the 7th number to Class 1.

The factors of the 8th number belong however to a new class i.e. Class 3, where 1 or more factors occur at most 3 times. On this occasion only one factor i.e. 2 occurs 3 times.

The proportion of all numbers belonging to Class 3  = 1/ζ(4) – 1/ζ(3) = .092 (approx) i.e. 9.2% of the accumulated total of factors for all numbers.

Then the factors of the next 7 numbers belong to Classes 1, 2, 1, 2, 1, 1 and 1 respectively.

Finally, the factors of the 16th number belongs to a new class i.e. Class 4, where 1 or more primes occur at most 4 times. (Here, 2 occurs 4 times!)

The proportion of all numbers belonging to Class 4 =  1/ζ(5) – 1/ζ(4) = .040 (approx) i.e. 4.0%.

Of course the factors of other numbers may belong with less frequency to other classes (that do not arise in this example).

In general, Class n can be defined as where 1 or more prime factors of a number occur at most n times.

And the probability of such occurrence =  1/{ζ(n + 1) – 1/ζ(n)}.

What is illustrated above refers to - what I term - the external nature of the number system, Here the proportion of all Classes - as  defined by the Riemann zeta function (for the positive integers) - relates to the overall frequency with which the natural numbers (as defined by prime factors of a certain type above) occur.

However there is a corresponding internal nature to the number system which - properly understood - is dynamically complementary with the external aspect. This relates to the corresponding proportion with respect to the number system as a whole with which the average frequency of the combined number of factors (associated with each Class Type) occurs.

So for example, we would intuitively expect on average to find more prime factors belonging to those classes i.e. 2 or higher (where one or more factors occur more than once), than Class 1 (where no such repetition of factors occurs).
The important implication here is that neither external nor internal aspects are independent of each other but rather synchronistically arise in a holistic manner that - ultimately - is of a purely relative nature.

So the internal aspect of measurement relates to calculation of the average frequency (with respect to the overall average frequency of prime factors for the number system as a whole).

Again, for convenience, we will refer to this category of numbers (which of course represents all the natural numbers) as Class 0.

Therefore, the task in terms of the internal distribution of the prime factors of numbers is to assess the average proportion of prime factors in each class (i.e. Class 1, Class 2, Class 3,......Class n) with respect to the average proportion of prime factors for the number system as a whole (i.e. Class 0).

Now, given acceptance with respect to the conjecture of the last entry that the total number of prime factors belonging to Class 1 represents 50%  of the factors of Class 0 (i.e. the cumulative sum of factors of all numbers), the total number of prime factors belonging to Class 2 represents 25% of all factors, the total number of prime factors belonging to Class 3 represents 12.5% of all factors and so on, we thereby have a ready means of calculating this internal distribution of prime factors.

For example though Class 1 accounts for 1/ζ(2) – 1/ζ(1) = .608 (approx) of the frequency of all numbers with no repeating prime factors, it accounts, as conjectured yesterday, for only 50% of the cumulative sum of all such factors (with respect to the number system as a whole).

Therefore to obtain the average frequency of such factors (with respect to the average frequency of factors for the number system as a whole) we divide .5 by 1/ζ(2) – 1/ζ(1) to obtain 1/2{ζ(2) – 1/ζ(1)} = .822 (approx).

Then to obtain the average frequency of prime factors of Class 2 - where 1 or more primes occur at most twice - we divide .25 by 1/ζ(3) – 1/ζ(2) = .224 (approx) to obtain 1/4{ζ(3) – 1/ζ(2)} = 1.116 (approx). So the average frequency of prime factors for this class is somewhat greater than for the number system as a whole.

Likewise to obtain the average frequency of prime factors of Class 3 - where 1 or more primes occur at most 3 times - we divide .125 by 1/ζ(4) – 1/ζ(3) = .092 (approx) to obtain 1/8{ζ(4) – 1/ζ(3)} = 1.358 (approx). So the average frequency of prime factors for this class as indeed all subsequent classes has further increased, and therefore again greater than for the number system as a whole.

In fact we can provide a simple general means of calculating all these proportions.

So to obtain the average frequency of prime factors of Class n - where 1 or more primes occur at most n times - we divide 1/2n by 1/{ζ(n + 1) – 1/ζ(n)} to obtain
1/2n{ζ(n + 1) – 1/ζ(n)}.