As is well
known the average spread or gap between primes (to n) is measured approximately
by log n.

For example
when n = 100, log n = 4.605…, so we would expect the average gap between primes
in the region of 100 to lie somewhere between 4 and 5 (approximately).

This
relative measurement will then steadily improve in accuracy for larger values
of n.

Therefore
the change in the spread between primes as the natural number increases by 1 to
n is given as log n – log (n – 1) = log n/(n – 1).

This
difference of logs is given by the infinite expression,

i.e. log n
– log (n – 1) = 1/n + 1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4}+……
For large n
this entails that log n – log (n – 1) is approximated very accurately by 1/n.

For example
if n = 1,000, then log n – log (n – 1) = log 1,000 – log 999

=
6.9077552… – 6.9067547… = .0010005…

And this
result - for a still comparatively low value of n - is already very close to
the reciprocal of n, i.e. 1/1000 = .001.

Indeed if
we take for greater accuracy the midpoint of the two numbers 999 and 1000, i.e.
999.5, then the result from subtracting the two log expressions is extremely
close to the reciprocal of 999.5, i.e. .0010000500025…

What we
have illustrated here stands here as perhaps the most remarkable - yet simple -
example of the nature of prime behaviour in that the change in the average
spread (or gap) as between successive prime numbers at n is intimately related
to the corresponding reciprocal of n.

In truth, the average spread between primes is better approximated by log n – 1 and this
distinction would be indeed significant over the lower ranges of n!

For example
there are 78,498 primes less than 1,000,000.

Now using
log n as the estimate for the average spread as between primes, this would
indicate using the formula n/log n, a total of 72,382 primes up to 1,000,000.

However
using log n – 1 as the average spread between primes this would give, using the
corresponding formula, n/ log n – 1, the much more accurate total of 78,030
primes (to 1,000,000).

However for
very large n, the distinction as between log n and log n – 1 becomes
increasingly less significant in relative terms.

What is not
commonly realised is that log n has an equally important complementary
significance with respect to the natural factors of a number.

Once, again
these natural factors are defined solely with respect to the composite natural
numbers and relate to all the factors of such a number (except 1).

So for
example, 12 contains 5 natural factors i.e. 2, 3, 4, 6 and 12!

Now, log n
- though once again log n – 1 is the better estimate for smaller values of n - measures
the average number of natural factors per unit to n.

Therefore, for example when n = 100, we would say that the average number of factors per
unit = 4.605…

This would
therefore suggest a cumulative total of 461 natural factors to 100.

However,
because n is still very small, as we have seen, log n – 1, would - relatively - provide the better estimate.

As log 100
– 1= 3.605…,, this would give a cumulative estimate of factors up to 100 of
361.

Indeed this
is very close to the actual cumulative total of factors to 100, which is 357!

Therefore
we can equally say that the change in the average frequency of natural factors
per unit as n increases by 1 is given by the reciprocal of n = 1/n.

So once
again, as we have seen, when for example, n = 1,000, the change in the average gap between
primes (per unit) is approximated by 1/1,000, likewise when n = 1,000, the
average change in the frequency of natural factors (per unit) is likewise
approximated by 1/1,000.

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