Monday, September 15, 2014

Interesting Log Relationships (1)

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 + 1/2n2 + 1/3n3 + 1/4n4 +……

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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