Now zooming out by a factor of 50, we get the above graph. Senior Max Planck
Institute mathematician Don Zagier, in
his 1977 article "The first 50 million primes" states:
"For me, the smoothness with which this curve climbs is one of the most astonishing facts in mathematics."(Note however that you are not looking at a smooth curve. Sufficiently powerful magnification would reveal that it was made of unit steps. The smoothness to which Zagier refers is smoothness in limit.) The juxtaposition of this property with the apparent 'randomness' of the individual positions of the primes creates a sort of tension which can be witnessed in many popular-mathematical accounts of the distribution of prime numbers. Adjectives such as "surprising", "astonishing", "remarkable", "striking", "beautiful", "stunning" and "breathtaking" have been used. Zagier captures this tension perfectly in the same article: "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers...grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision." |