In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. The prime numbers "thin out" as one looks at larger and larger numbers, and the prime number theorem gives a precise description of exactly how much they thin out. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

Roughly speaking, the prime number theorem states that if you randomly select a number nearby some large number N, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime. The word random is used to express lack of order, purpose, cause, or predictability in non-scientific parlance. ... Probability is the chance that something is likely to happen or be the case. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...

Statement of the theorem

Graph comparing π(x), x / ln x and Li(x)

Let π(x) be the prime counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1, which is expressed by the formula Image File history File links PrimeNumberTheorem. ... Image File history File links PrimeNumberTheorem. ... In mathematics, the prime counting function is the function counting the number of primes less than or equal to some real number x. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

known as the asymptotic law of distribution of prime numbers. Using Landau notation this result can be restated as It has been suggested that this article or section be merged into Big O notation. ...

.

This notation (and the theorem) does not say anything about the limit of the difference of the two functions as x approaches infinity. (Indeed, the behavior of this difference is very complicated and related to the Riemann hypothesis.) Instead, the theorem states that x/ln(x) approximates π(x) in the sense that the relative error of this approximation approaches 0 as x approaches infinity. Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... In the mathematical subfield of numerical analysis the approximation error in some data is the difference between the exact value and the value used. ...

The prime number theorem is equivalent to the statement that the nth prime number p_n is approximately equal to n ln(n), again with the relative error of this approximation approaching 0 as n approaches infinity.

History of the asymptotic law of distribution of prime numbers and its proof

Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1796 that π(x) is approximated well by the function x/(ln(x)-B), where B=1.08... is a certain constant close to 1. Carl Friedrich Gauss considered the same question, and based on the computational evidence available to him and on some heuristic reasoning, he came up with his own approximating function, the logarithmic integral li(x), although he did not publish his results. Both Legendre's and Gauss's formulas imply the same conjectured asymptotic equivalence of π(x) and x / ln(x) stated above, although it turned out that Gauss's approximation is considerably better if one considers the differences instead of quotients. Anton Felkel as depicted in the fronstspice onf his 1761 Tables. ... Baron Jurij Bartolomej Vega (also correct Veha; official Latin Georgius Bartholomaei Vecha; German Georg Freiherr von Vega) (March 23, 1754 – September 26, 1802) was a Slovenian mathematician, physicist and artillery officer. ... Adrien-Marie Legendre (September 18, 1752 – January 10, 1833) was a French mathematician. ... Johann Carl Friedrich Gauß (in English literature sometimes written Gauss) ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... In mathematics, the logarithmic integral or integral logarithm li(x) is a non-elementary function defined for all positive real numbers x≠ 1 by the definite integral: Here, ln denotes the natural logarithm. ...

In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity exists at all, then it is necessarily equal to one. He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all x. Although Chebyshev's paper did not quite prove the Prime Number Theorem, he used his estimates for π(x) to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2. Pafnuty Lvovich Chebyshev (Russian: ) ( May 26 [O.S. May 14] 1821 – December 8 [O.S. November 26] 1894) was a Russian mathematician. ... Bertrands postulate states that if n > 3 is an integer, then there always exists at least one prime number p with n < p < 2n âˆ’ 2. ...

Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject. Riemann introduced revolutionary ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending these deep ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Hadamard and de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.^[1] On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... This page is a candidate for speedy deletion. ... Charles-Jean de la Vallée-Poussin (August 14, 1866 - March 2, 1962) was a Belgian mathematician. ...

During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdös (1949). Atle Selberg (born June 17, 1917) is a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. ... Paul ErdÅ‘s Paul ErdÅ‘s (March 26, 1913 – September 20, 1996) was an immensely prolific and famously eccentric mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory. ...

A very rough proof sketch

In a lecture on prime numbers for a general audience, Fields medalist Terry Tao described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes. We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs discussed below^[2]. The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... Terence Chi-Shen Tao ( 陶哲軒)(born 1975) is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory. ... The von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ...

The prime counting function in terms of the logarithmic integral

Carl Friedrich Gauss conjectured that an even better approximation to π(x) is given by the offset logarithmic integral function Li(x), defined by Johann Carl Friedrich Gauß (in English literature sometimes written Gauss) ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... The offset logarithmic integral, or European logarithmic integral, is a non-elementary function Li(x) differing by a constant from the logarithmic integral function li(x), defined such that: Explicitly, this means where ln is the natural logarithm. ...

Indeed, this integral is strongly suggestive of the notion that the 'density' of primes around t should be 1/lnt. This function is related to the logarithm by the asymptotic expansion In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular...

So, the prime number theorem can also be written as π(x) ~ Li(x). In fact, it follows from the proof of Hadamard and de la Vallée Poussin that

for some positive constant a, where O(…) is the big O notation. This has been improved to Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. ...

Because of the connection between the Riemann zeta function and π(x), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today. More specifically, Helge von Koch showed in 1901^[3] that, if and only if the Riemann hypothesis is true, the error term in the above relation can be improved to Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Niels Fabian Helge von Koch (January 25, 1870 - March 11, 1924) was a Swedish mathematician, who gave his name to the famous fractal known as the Koch curve, which was one of the earliest fractal curves to have been described. ... This article does not cite any references or sources. ...

The constant involved in the big O notation was estimated in 1976 by Lowell Schoenfeld^[4]: assuming the Riemann hypothesis, Lowell Schoenfeld (1920-2002) was an American mathematician known for his work in analytic number theory. ...

for all x ≥ 2657. He also derived a similar bound for the Chebyshev prime counting function ψ: The Chebyshev function , with The Chebyshev function , for The Chebyshev function , for The Chebyshev function is either of two related functions. ...

for all x ≥ 73.2.

The logarithmic integral Li(x) is larger than π(x) for "small" values of x. However, in 1914, J. E. Littlewood proved that this is not always the case. The first value of x where π(x) exceeds Li(x) is around x = 10³¹⁶; see the article on Skewes' number for more details. John Edensor Littlewood (June 9, 1885 – September 6, 1977) was a British mathematician. ... In number theory, Skewes number is by definition the smallest natural number x for which Ï€(x) − Li(x) ≥ 0 where Ï€(x) is the prime counting function and Li(x) is the offset logarithmic integral. ...

Elementary proofs

In the first half of the twentieth century, some mathematicians felt that there exists a hierarchy of techniques in mathematics, and that the prime number theorem is a "deep" theorem, whose proof requires complex analysis. Methods with only real variables were supposed to be inadequate. G. H. Hardy was one notable member of this group[1]. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ...

The formulation of this belief was somewhat shaken by a proof of the prime number theorem based on Wiener's tauberian theorem, though this could be circumvented by awarding Wiener's theorem "depth" itself equivalent to the complex methods. The notion of "elementary proof" in number theory is not usually defined precisely, but it usually seems to correspond roughly to proofs that can be carried out in Peano arithmetic, rather than more powerful theories, such as second order arithmetic. There are statements of Peano arithmetic that can be proved in second order arithmetic but not first order arithmetic (see the Paris-Harrington theorem for an example), but they seem in practice to be rare. However, Atle Selberg found an elementary proof of the prime number theorem in 1949, which uses only number-theoretic means. (Paul Erdős used Selberg's ideas to produce a slightly different elementary proof at about the same time.) Selberg's work effectively laid rest to the whole concept of "depth" for the prime number theorem, showing that technically "elementary" methods (in other words Peano arithmetic) were sharper than previously expected. In 2001 Sudac showed that the prime number theorem can even be proved in primitive recursive arithmetic^[5], a much weaker theory than Peano arithmetic. In mathematics, Wieners tauberian theorem is a 1932 result of Norbert Wiener. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... In mathematical logic, second order arithmetic is a stronger version of Peano arithmetic that allows quantification over subsets of the integers, rather than just over integers. ... In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory is true but not provable in Peano arithmetic. ... Atle Selberg (born June 17, 1917) is a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. ... In mathematics a proof is said to be elementary if uses only ideas from within its field and closely related issues. ... Paul ErdÅ‘s, also Pál ErdÅ‘s, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory... Primitive recursive arithmetic, or PRA, is a formal system of quantifier-free, or zeroth-order arithmetic, and is considered to be the basic arithmetic of finitism. ...

Avigad et al. wrote a computer verified version of this elementary proof in the Isabelle theorem prover in 2005^[6]. The Isabelle theorem prover is an interactive theorem proving framework, a successor of the HOL theorem prover. ...

The prime number theorem for arithmetic progressions

Let π_n,a(x) denote the number of primes in the arithmetic progression a, a + n, a + 2n, a + 3n, … less than x. Dirichlet and Legendre conjectured, and Vallée Poussin proved, that, if a and n are coprime, then In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Adrien-Marie Legendre (September 18, 1752&#8211;January 10, 1833) was a French mathematician. ... Coprime - Wikipedia /**/ @import /skins-1. ...

where φ(·) is the Euler's totient function. In other words, the primes are distributed evenly among the residue classes [a] modulo n with gcd (a, n) = 1. This can been proved using similar methods used by Newman for his proof of the prime number theorem^[7]. The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

Littlewood showed in 1914 that there are infinitely many sign changes for the function :π_1,4(x) − π_3,4(x).

Bounds on the prime counting function

The prime number theorem is an asymptotic result. Hence, it cannot be used to bound π(x).

However, some bounds on π(x) are known, for instance Pierre Dusart's Pierre Dusart is a French mathematician who specializes in number theory. ...

The first inequality holds for all x ≥ 599 and the second one for x ≥ 355991^[8].

A weaker but sometimes useful bound is

for x ≥ 55^[9]. In Dusart's thesis you also find slightly stronger versions of this type of inequality (valid for larger x.)

Approximations for the nth prime number

As a consequence of the prime number theorem, one gets an asymptotic expression for the nth prime number, denoted by p_n: In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

A better approximation is

^[10]

Rosser's theorem states that p_n is larger than n ln n. This can be improved by the following pair of bounds^[11]: In mathematics, Rossers theorem was proved by J. Barkley Rosser in 1938. ...

The left inequality is due to Pierre Dusart^[12] and is valid for n ≥ 2. The right inequality is due to Rosser.

Table of π(x), x / ln x, and Li(x)

The table compares exact values of π(x) to the two approximations x / ln x and Li(x). The last column, x / π(x), is the average prime gap below x. The n-th prime gap (short for prime number gap), denoted gn, is the difference between the n+1-th and n-th prime number, pn. ...

x	π(x)[13]	π(x) − x / ln x[14]	π(x) / (x / ln x)	Li(x) − π(x)[15]	x / π(x)
10	4	−0.3	0.921	2.2	2.500
102	25	3.3	1.151	5.1	4.000
103	168	23	1.161	10	5.952
104	1,229	143	1.132	17	8.137
105	9,592	906	1.104	38	10.425
106	78,498	6,116	1.084	130	12.740
107	664,579	44,158	1.071	339	15.047
108	5,761,455	332,774	1.061	754	17.357
109	50,847,534	2,592,592	1.054	1,701	19.667
1010	455,052,511	20,758,029	1.048	3,104	21.975
1011	4,118,054,813	169,923,159	1.043	11,588	24.283
1012	37,607,912,018	1,416,705,193	1.039	38,263	26.590
1013	346,065,536,839	11,992,858,452	1.034	108,971	28.896
1014	3,204,941,750,802	102,838,308,636	1.033	314,890	31.202
1015	29,844,570,422,669	891,604,962,452	1.031	1,052,619	33.507
1016	279,238,341,033,925	7,804,289,844,393	1.029	3,214,632	35.812
1017	2,623,557,157,654,233	68,883,734,693,281	1.027	7,956,589	38.116
1018	24,739,954,287,740,860	612,483,070,893,536	1.025	21,949,555	40.420
1019	234,057,667,276,344,607	5,481,624,169,369,960	1.024	99,877,775	42.725
1020	2,220,819,602,560,918,840	49,347,193,044,659,701	1.023	222,744,644	45.028
1021	21,127,269,486,018,731,928	446,579,871,578,168,707	1.022	597,394,254	47.332
1022	201,467,286,689,315,906,290	4,060,704,006,019,620,994	1.021	1,932,355,208	49.636
1023	1,925,320,391,606,818,006,727	37,083,513,766,592,669,113	1.020	7,236,148,412	51.939

Analogue for irreducible polynomials over a finite field

There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem. In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...

To state it precisely, let F = GF(q) be the finite field with q elements, for some fixed q, and let N_n be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients chosen from F, which cannot be written as products of polynomials of smaller degree. In this setting, these polynomials play the role of the prime numbers, since all other monic polynomials are built up of products of them. One can then prove that In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ...

If we make the substitution x = qⁿ, then the right hand side is just

which makes the analogy clearer. Since there are precisely qⁿ monic polynomials of degree n (including the reducible ones), this can be rephrased as follows: if you select a monic polynomial of degree n randomly, then the probability of it being irreducible is about 1/n.

One can even prove an analogue of the Riemann hypothesis, namely that

The proofs of these statements are far simpler than in the classical case. It involves a short combinatorial argument, summarised as follows. Every element of the degree n extension of F is a root of some irreducible polynomial whose degree d divides n; by counting these roots in two different ways one establishes that

where the sum is over all divisors d of n. Möbius inversion then yields In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. ...

where μ(k) is the Möbius function. (This formula was known to Gauss.) The main term occurs for d = n, and it is not difficult to bound the remaining terms. The "Riemann hypothesis" statement depends on the fact that the largest proper divisor of n can be no larger than n/2. The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

See also

• Abstract analytic number theory for information about generalizations of the theorem.
• Landau prime ideal theorem for a generalization to prime ideals in algebraic number fields.
• Prime gap

Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. ... In mathematics, the prime ideal theorem of algebraic number theory is the number field generalization of the prime number theorem. ... The n-th prime gap (short for prime number gap), denoted gn, is the difference between the n+1-th and n-th prime number, pn. ...

References

1. ^ Ingham, A.E. (1990). The Distribution of Prime Numbers. Cambridge University Press, 2-5. ISBN 0-521-39789-8. 
2. ^ Video and slides of Tao's lecture on primes, UCLA January 2007.
3. ^ Helge von Koch (Dec 1901). "Sur la distribution des nombres premiers". Acta Mathematica 24 (1): 159-182. 
4. ^ Lowell Schoenfeld (Apr 1976). "Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x), II". Mathematics of Computation 30 (134): 337–360. 
5. ^ Olivier Sudac (Apr 2001). "The prime number theorem is PRA-provable". Theoretical Computer Science 257 (1–2): 185-239. 
6. ^ Jeremy Avigad, Kevin Donnelly, David Gray, Paul Raff (2005). "A formally verified proof of the prime number theorem". e-print cs. AI/0509025 in the ArXiv. 
7. ^ Ivan Soprounov (1998). "A short proof of the Prime Number Theorem for arithmetic progressions". 
8. ^ Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers, doctoral thesis for l'Université de Limoges (1998).
9. ^ Barkley Rosser (Jan 1941). "Explicit Bounds for Some Functions of Prime Numbers". American Journal of Mathematics 63 (1): 211-232. 
10. ^ Michele Cipolla (1902). "La determinazione assintotica dell'n^imo numero primo". Matematiche Napoli 3: 132-166. 
11. ^ Eric Bach, Jeffrey Shallit (1996). Algorithmic Number Theory. MIT Press. ISBN 0-262-02405-5. 
12. ^ Pierre Dusart (1999). "The kth prime is greater than k(ln k + ln ln k-1) for k>=2". Mathematics of Computation 68. 
13. ^ Number of primes < 10^n (A006880). On-Line Encyclopedia of Integer Sequences.
14. ^ Difference between pi(10^n) and the integer nearest to 10^n / log(10^n) (A057835). On-Line Encyclopedia of Integer Sequences.
15. ^ Difference between Li(10^n) and Pi(10^n), where Li(x) = integral of log(x) and Pi(x) = number of primes <= x (A057752). On-Line Encyclopedia of Integer Sequences.

• G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp.119-196.
• Andrew Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal, vol. 1, pages 12–28, 1995.

arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Eric Bach is an American computer scientist who has made contributions to computational number theory. ... Jeffrey Outlaw Shallit (born 1957) is a computer scientist, number theorist, and a noted advocate for civil liberties on the Internet. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

External links

• Table of Primes by Anton Felkel.
• Prime formulas and Prime number theorem at MathWorld.
• Prime number theorem on PlanetMath
• How Many Primes Are There? and The Gaps between Primes by Chris Caldwell, University of Tennessee at Martin.
• Tables of prime counting functions by Tomás Oliveira e Silva