Riemann Zeta Function: Important Number Relationships (11)

To illustrate the subject matter of the previous blog entry "Riemann Zeta Function: Important Number Relations (10)", I am now providing some preliminary sample evidence that I conducted in assessing the ratios mentioned yesterday.

In this sample, I surveyed the 500 numbers of the 14 digit numbers in sequence from 15000000006001 - 150000000006500 with respect to frequency and also with respect to the combined number of factors contained.
Thus for example with reference to the first 100 of these numbers, 61 represents the frequency of those numbers which contain at most 1 prime factor (i.e. no repeating primes) and 215 then represents the cumulative number of prime factors contained with respect to these 61 numbers!

So in the case of each prime factor repeating category, I then worked out both the total frequency for the 500 numbers and the corresponding total cumulative frequency of the prime factors involved.

All Nos. Fr. Factors	At Most 1 Fr.  Factors	At Most 2 Fr.  Factors	At Most 3 Fr.  Factors	At Most 4 Fr.  Factors
100         456	60           215	22         119	8              55	4            27
100         451	61           207	22         119	9              55	3            20
100         453	60           204	23         128	8              53	4            34
100         464	63           227	17           89	10             69	4            30
100         454	59           203	24         132	9              58	4            30
500         2278	303         1056	108        587	44            290	19          142

   Prop.          1               .4635                   .2576               .1273                .0623

     Av.       4.556             3.485                  5.435                6.591               7.474  

I then worked out the average no. of factors involved in each case, which then provides estimates for the ratios involved (as mentioned in the last entry).

For example, the ratio here of the average factor frequency of those numbers with at most 1 prime factor to those where at most 1 or more factors occur 2 times = 3.485/5.435 =  .641.

The suggested estimate directly based on the Riemann zeta function (for positive integers) of 1/ζ(2) = .6079. So the estimate is not that accurate. However our sample is relatively small with the average number of factors per number involved still at a very low rate!

The next ratio estimate of the corresponding average factor frequency of those numbers where at most 1 or more factors occur 2 times to those where at most 1 or more prime factors occur 3 times = 5.435/6.591 = .825.

In this case, the estimate is very close to the true postulated value i.e.  1/ζ(3) = .832, though not too much significance should be read into this fact.

Then the corresponding estimate of the ratio of average factor frequency of  those numbers where at most 1 or more prime factors occur 4 times = 6.591/7.474 = .882. This compares with the suggested result of 1ζ(4) = .924. Again given the limited data on which the sample estimate is based, this result gives support to the view that the underlying pattern here is indeed described by the Riemann zeta function for 1/ζ(2), 1/ζ(3), 1/ζ(4) and so on! 

However as I made further estimates, I became aware of an important trend that eventually led me to abandon my initial estimates.

Basically, what I found is that these estimates of ratios tended to vary depending on how high up the number scale numbers (with their constituent prime factors) are taken. For example, the important first ratio (measuring the average factor frequency of those numbers where at most factors occur once (i.e. where all are non-repeating) to those with at most 1 or more factors occur 2 times tended to increase for numbers higher up the number scale.

While still confident that a relationship entailing the Riemann zeta function (for positive integer values) still governed the internal distribution of the primes, this led me to look in another direction (which I return to in the next entry).

This direction came from the additional consideration of the the total number of accumulated prime factors associated with each of our categories in the number system.

This entails combining distributions with respect to both the external and internal aspects of the number system.

For example though 61% (approx) of all numbers are externally composed of prime factor combinations (where no prime occurs more than once) the internal average frequency of overall factor occurrence for such numbers is less than any of the other categories.

Thus combining both external and internal considerations - in the line marked as prop. (i.e. proportion) - the proportion of all accumulated prime factors (2278 in our example) in relation to the total  = 1.

The with respect to the accumulation of prime factors (for those numbers where each factor occurs at most 1 time (= 1056), the proportion  - in our example - is slightly less than 1/2 of total (i.e. .4635).

Then with relation to those numbers where at most 1 or more prime factors occur 2 times (= 587), the proportion with respect to the total accumulated prime factors is very close to 1/4 (i.e. .2576).

Next, in relation to those numbers where at most 1 or more prime factors occur 3 times (= 290), the proportion is very close to 1/8 (i.e. .1273).

Then finally (in our example) in relation to those numbers where at most 1 or more prime factors occur 4 times (= 142), the proportion is very close to 1/16 (i.e. .0623). 

Then bearing in mind the nature of the Riemann zeta function (for positive integers > 1) and the need to ensure that the overall sum of percentages (for all prime factor combinations) = 100%, this would imply that these successive proportions would closely follow the series 1 + 1/2 + 1/4 + 1/8 + 1/16 .....

Now the only estimate in or sample that is not very close to the postulated proportion is that for the accumulated factors of those numbers where no prime occurs more than once.

Once again our estimated value here is .4635 whereas the postulated value is .5.

However there is reason to believe that this proportion steadily rises towards .5 as we move up the number scale.

For example when I earlier measured this proportion with respect to the 500 numbers (200,001 - 200,500), the estimate for this proportion (i.e. ratio of accumulated number of factors of those numbers where no prime factor occurs more than 1, with respect to accumulation of prime factors for all numbers in this range) the estimated value was less than .45.

Then from various other estimates, I found that this proportion tended to steadily rise as we move higher up the number scale.

For example, I measured this proportion for the 1000 16 digit numbers from 800000000000001 - 800000000001000 and obtained the value .4688.

Now this might appear but a small increase (with respect to the 14 digit number estimate of .4635), but one must remember that the average number of prime factors per number changes at a very slow rate. Therefore to obtain numbers with an appreciably higher number of prime factors we would need to move significantly higher up the number scale and unfortunately way beyond the range for which prime factors are readily obtainable!

However in view of the fact that the total proportion is necessarily equal to 1 and likewise in view of the fact that subsequent proportions (for numbers where 1 or prime factors occur more than once) conform so closely to the series 1/4 + 1/8/ + 1/16 + ....., then it is eminently reasonable to assume that the proportion corresponding to the first term ultimately → 1/2 for numbers sufficiently high up the number scale. 

So, in general, the proportion of the total accumulated frequency of all factors for those numbers where at most 1 or more prime factors occur n times → 1/2ⁿ (with the accuracy of this steadily improving as the number increases).