from
*What is Mathematics?* by R. Courant and H. Robbins
Near the end of the book, we find a section with this name. The
authors state:
*"By a procedure typical of...statistical mechanics we...
*[*make*]*
plausible the...law of the distribution of primes." *
Two results are combined:
which was proven earlier in the book, by means of a fairly straightforward
argument, and
which is proven in this section, using a simple counting argument,
to give an elementary result with an elementary proof:
Note that the RHS is a simple step function based on the positions
of the primes. The above result states that it is asymptotic to the
natural logarithm function.
Here we see the two functions in the range [0,100]:
Here we see the two functions in the range [0,1000]:
This relationship between the logarithm function and the
distribution of primes is possibly the best "suggestive" evidence for
the prime number theorem
The authors assume the existence of a density function
*w*(*x*) for the primes, with certain basic properties,
and then rewrite
in terms of an integral involving
this density. Elementary calculus methods are then used to deduce
that
*w*(*x*) ~ (*x* - 1)/*x*log *x*,
leading to the prime number theorem.
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