Prime number theorem

The prime number theorem allows an estimation of the distribution of prime numbers using logarithms. The relationship between primes and logarithms was already on the 15 -year-old Carl Friedrich Gauss in 1793 and independently of him by Adrien -Marie Legendre in 1798 suspected but until 1896 proved independently by Jacques Salomon Hadamard and Charles Jean de la Vallée Poussin.

The prime function

Furthermore, the prime function is defined for arbitrary real numbers as the number of prime numbers that are no greater than. Formally, one can write:

Here, the symbol denotes the set of prime numbers, the notation is the number of elements of the set

The prime number theorem

The prime number theorem states:

If we call two real functions and asymptotically equivalent if the ratio for converges to 1, it can be the prime number theorem also formulated as follows:

The functions and are asymptotically equivalent.

Stronger forms of the prime number theorem

Better approximations as supplied by the so-called logarithmic integral, as the

Is defined. ( The integral representation for Li (x ) is chosen because the primitives of 1/ln (x ) are not elementary. )

The logarithmic integral is asymptotically equivalent to including to

One can even show:

This is a positive constant a Landau - symbol, i.e., there is a constant such that

Applies to all.

Assuming the Riemann hypothesis, and only this, one can estimate the error to

Improve.

History

Adrien -Marie Legendre published in 1798 as the first in his Théorie des nombres (Treatise on Number Theory ) regardless of Gauss the suspected relationship between primes and logarithms. In the second edition of this work in 1808, he improved the estimate of approximately equal to

A first step towards a proof was Pafnuti Lvovitch Chebyshev, who showed in 1851 the following weaker form of the prime number theorem:

For all sufficiently large That is, the number of primes at a given size by no more than about 10 % above or below is different from the logarithmic function. The English mathematician James Joseph Sylvester, then a professor at the Johns Hopkins University in Baltimore, refined 1892 Tschebyschows method and showed that for the inequality for sufficiently large x satisfies the lower limit and the upper limit 0.95695 1.04423, the deviation ie a maximum only is about 5%.

In his famous work on the number of primes below a given size (1859 ) Bernhard Riemann has shown the relationship between the distribution of prime numbers and the properties of the Riemann zeta function. The German mathematician Hans von Mangoldt 1895 proved the main result of the Riemann work that the prime number theorem, the set is equivalent to the Riemann zeta function has no zeros with real part 1. Both Hadamard and de la Vallée Poussin in 1896 have demonstrated the non-existence of such zeros. Their proofs of the prime number theorem are not " elementary ", but using function-theoretic methods. For many years was considered an elementary proof of the prime number theorem for impossible, which was established in 1949 refuted by the evidence found by Atle Selberg and Paul Erdős (and it by no means " easy " means " elemental "). Later, numerous variants and simplifications were still found evidence of this.

Numerical examples

The size is called prime density.

Comparing with the values ​​of the table, it seems as if it would always apply. In fact, the difference becomes larger changes the sign infinitely often as JE Littlewood showed in 1914. Thus, the Gaussian formula underestimates the number of primes in a sufficiently large range of numbers, the Stanley Skewes 1933 was estimated by the eponymous Skewes number upwards. Russell Sherman Lehman introduced in 1966 to an important theorem about the upper limit and could press " manageable " to a size of 1.165 · 101165. Using the Lehman 's theorem succeeded the Dutch mathematician Herman te Riele to show in 1986 that there are more than 10180 consecutive numbers x are between 6,627 and 6,687 · 10370 · 10370 applies. Currently offers the most lowest value, it is also based on the results of Lehman, determined in 2000, the two mathematicians Carter Bays and Richard Hudson, who showed that such a proven Littlewood change in the area of 1.39822 10316 · (probably) for the first time occurs.