By Stephen P. Richards
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Extra resources for A Number for Your Thoughts
Quite obviously not all tables of numbers behave in this way. If, for example, we look at a set of log tables themselves (which record the logarithms of all numbers between 1000 and 9999) then we find that this particular collection of numbers is quite different, being overly weighted at the 9 end of the spectrum. Also it is clear that if the list contains properties with only a small scatter about their average, such as for instance the annual rainfall where you live, then the first digit distribution will be very different from Benford's finding.
And was finally proved rigorously many years later. Its importance is that it enables us to predict the number of primes less than some large number n with considerable accuracy without the need to find them all and count them explicitly. The result does not mean that the difference between 1r(n) and the function n/ln(n) is necessarily small in an absolute sense, but only that it is small compared to the number 1r(n) itself. In more recent years more complicated functions have been found which approximate the actual prime number distribution even more closely.
It is evident from the few examples already given that it certainly does not increase steadily with increasing values of n. Moreover, since the prime numbers go on forever, and since p (n) is equal to 1 for all prime numbers, there must be some values of p (n) equal to 1 no matter how large n becomes. On the other hand, the value of p (n) for other numbers can grow much larger as n increases. Indeed p (n) can become extremely large for some n-values because, if we think back to that number which we called factorial n' and wrote n!
A Number for Your Thoughts by Stephen P. Richards