3. Distribution of the primes
This small sample of data is typical of the complex behavior of the "distribution" of primes. The hunch that as one gets further out in the integers the primes are less "dense" and the appearance of twin primes becomes more rare, is borne out. However, we see from the table that the distribution of primes is very complex. Yet mathematicians have remarkable results about the distribution of the primes.
What seemed to be the case was that as x got larger, the ratio between π(x) and x/(ln x) got closer to 1. This result is known today as the Prime Number Theorem. To provide two data points, when x is a million, x/(ln x) is about 72,382, while π(x) is 78,498, and for a billion the corresponding values are 48,254,942 and 50,847,534. Thus,x/ln(x) gives an increasingly good estimate for π(x). For a billion, the estimate is off by about 5 percent.
ax/(ln x) < π(x) < bx/(ln x).
However, the Prime Number Theorem was first proved independently in 1896 by the French mathematician Jacques Hadamard (1865-1962) and the Belgian mathematician Charles de la Vallée Poussin (1866-1962). (Not only did these men provide independent proofs, but they were born a year apart and died in the same year.) The proof techniques that were used involved the theory of functions of a complex variable. Eventually, a proof (1949) of the Prime Number Theorem which did not involve the use of the theory of complex functions was given by Paul Erdös and Atle Selberg. We see here a classic example of how long it sometimes takes to give a rigorous proof of a mathematical result which "empirically" was known for a long time. Still unresolved was the honing of powerful conceptual tools that made it possible to provide a proof and to get additional structural insights as well. Researchers are still trying to get new and different insights into the distribution of the primes.
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