"The question of the diminishing frequency of primes was the subject of much speculation before any definite results emerged. The problem assumed a much more precise form with the publication by Legendre in 1808 (after a less definite statement in 1798) of a remarkable empirical formula for the approximate representation of . Legendre asserted that, for large values of x, is approximately equal to

x/(log x - B)

where log x is the natural (Naperian) logarithm of x and B a certain numerical constant - a theorem described by Abel (in a letter written in 1823) as the 'most remarkable in the whole of mathematics'. A similar, though not identical, formula was proposed independently by Gauss. Gauss's method, which consisted in counting the primes in blocks of a thousand consecutive integers, suggested the function 1/log x as an approximation to the average density of distribution ('number of primes per unit interval') in the neighbourhood of a large number x...

Gauss's observations were communicated to Encke in 1849, and first published in 1863; but they appear to have commenced as early as 1791 when Gauss was fourteen years old...

The proposition, which is now known as the 'prime number theorem', is the central theorem in the theory of the distribution of primes. The problem of deciding its truth or falsehood engaged the attention of mathematicians for about a hundred years."

(A.E. Ingham, The Distribution of Prime Numbers)




Here the sequence of primes is presented graphically in terms of a step function or counting function which is traditionally denoted .

The height of the graph at horizontal position x indicates the number of primes less than or equal to x. Hence at each prime value of x we see a vertical jump of one unit.



By zooming out to see the distribution of primes within the first 100 natural numbers, we see that the discrete step function is beginning to suggest a curve.



Zooming out by another factor of 10, the suggested curve becomes even more apparent. Zooming much further, we would expect to see the "granular" nature of the actual graph vanish into the pixelation of the screen.


Now zooming out by a factor of 50, we get the above graph. Senior Max Planck Institute mathematician Don Zagier, in his article "The first 50 million primes" [Mathematical Intelligencer, 0 (1977) 1-19] 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. Zagier is using the term "smoothness" in a suggestive, nontechnical sense here.)

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."


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