Estimation of Frequency of Prime Numbers (2)

In a previous blog entry, I suggested a simple improvement to the simple log formula (n/log n) as a means of predicting the frequency of primes.

Here I set n₁ = n/log n and then obtained the new modified estimate,

i.e. n/log n + n₁/log n₁  

However an even simpler improvement - and ultimately more accurate estimate - is obtained though the slight modification of the original log formula i.e. n/(log n – 1).

So once again I provide a table in multiples of 10 up to 10,000,000,000 showing the actual occurrence of primes at each stage as against the predicted values using the original simple estimate (n/log n) and the new modified version n/(log n – 1). 

Up to n	Actual no.	n/log n	n/(log n – 1)
10	4	4	8
100	25	22	28
1000	168	145	169
10000	1229	1086	1218
100000	9592	8686	9512
1000000	78498	72382	78030
10000000	664579	620421	661459
100000000	5761455	5428681	5740304
1000000000	50847534	48254942	50701542
10000000000	455052511	434294482	454011971

It can be seen readily from the above table that n/(log n – 1) gives a much more accurate estimation of the frequency of primes than n/log n. Indeed in the final entry in the table for 10¹⁰, the accuracy of the first log formula is 95.44% whereas with the modified version it is 99.77%.

It is also apparent from the above that both formulae give an under estimate of the actual number of primes.

By contrast the earlier modified estimate, n/log n + n₁/log n₁, gives an over estimate.

This would suggest that a simple mean of the two estimates i.e. n/(log n – 1) and  n/log n + n₁/log n₁ would therefore give a more accurate estimate and indeed over the range of values in the table this is certainly the case with the actual value roughly the midpoint of the two modified estimates. However at higher values (and I tested to 10^25) a distinct bias enters in with the excess in the estimate given by n/log n + n₁/log n₁, becoming increasingly greater than the corresponding deficit in the estimate of n/(log n – 1).

There is an important reason why n/(log n  – 1) should prove  a better estimate than n/log n.

Log n gives the average spread or gap as between the primes. For example in the region of 1000 we would expect the average gap as between primes to approximate 7 so that the relative frequency = 1/7 (= 1/log n). Another way of expressing this is by saying that in the region of 1000 we would on average expect an unbroken sequence of 6 composite numbers before encountering a primes (i.e. log n  – 1) .

Now as I have stated before the primes (from the cardinal perspective) represent the independent aspect with respect to the number system (so that all other composite numbers are derived from the relationship between primes (as building blocks). 
So the composite numbers represent the interdependent aspect with respect to the number system. And in dynamic interactive terms, the relationship of primes expresses this complementary relationship as between independence and interdependence respectively.

Thus properly speaking the relationship of the primes is not directly with the overall number system (which includes primes) but rather with those composite members (that are qualitatively distinct from primes).   

The beauty of this modified formulation n/(log n – 1) is then not only in the fact that it serves as a much better estimator of the primes but that in turn it can be seen to bear a simple inverse relationship with the formula n(log n – 1).
This, as we have seen, accurately predicts the corresponding number of  factors with respect to the composite numbers (bearing in turn a very close relationship with the frequency of occurrence of the non-trivial zeros).

So therefore the very purpose of this blog entry (and so many before) is to show how the ability to look at the number system in a dynamic interactive manner can reveal many of its great secrets in a very simple form.