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我们会通过消息、邮箱等方式尽快将举报结果通知您。From Wikipedia, the free encyclopedia
In , the prime number theorem (PNT) describes the
distribution of the
among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by
in 1896 using ideas introduced by
(in particular, the ).
The first such distribution found is π(N) ~ N/log(N), where π(N) is the
and log(N) is the
of N. This means that for large enough N, the
that a random integer not greater than N is prime is very close to 1 / log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N).
Graph showing ratio of the prime-counting function π(x) to two of its approximations, x / log x and Li(x). As x increases (note x axis is logarithmic), both ratios tend towards 1. The ratio for x / log x converges from above very slowly, while the ratio for Li(x) converges more quickly from below.
Log-log plot showing absolute error of x / log x and Li(x), two approximations to the prime-counting function π(x). Unlike the ratio, the difference between π(x) and x / log x increases without bound as x increases. On the other hand, Li(x) - π(x) switches sign infinitely many times.
Let π(x) be the
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 x / log x is a good approximation to π(x), in the sense that the
of the quotient of the two functions π(x) and x / log x as x increases without bound is 1:
{\displaystyle \lim _{x\to \infty }{\frac {\;\pi (x)\;}{\frac {x}{\log(x)}}}=1,}
known as the asymptotic law of distribution of prime numbers. Using
this result can be restated as
{\displaystyle \pi (x)\sim {\frac {x}{\log x}}.}
This notation (and the ) does not say anything about the limit of the difference of the two functions as x increases without bound. Instead, the theorem states that x / log x approximates π(x) in the sense that the
of this approximation approaches 0 as x increases without bound.
The prime number theorem is equivalent to the statement that the nth prime number pn satisfies
{\displaystyle p_{n}\sim n\log(n),}
the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as n increases without bound. For example, the 2×1017th prime number is 8512677386048191063, and (2×1017)log(2×1017) rounds to 7967418752291744388, a relative error of about 6.4%.
The prime number theorem is also equivalent to
{\displaystyle \lim _{x\to \infty }{\frac {\vartheta (x)}{x}}=\lim _{x\to \infty }{\frac {\psi (x)}{x}}=1,}
where ? and ψ are
respectively.
Distribution of primes up to
(9699690).
Based on the tables by
conjectured in 1797 or 1798 that π(a) is approximated by the function a / (A log a + B), where A and B are unspecified constants. In the second edition of his book on number theory (1808) he then made a , with A = 1 and B = -1.08366.
considered the same question at age 15 or 16 "in the year 1792 or 1793", according to his own recollection in 1849. In 1838
came up with his own approximating function, the
li(x) (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(x) and x / log(x) stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
In two papers from 1848 and 1850, the Russian mathematician
attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s), for real values of the argument "s", as in works of , as early as 1737. Chebyshev's papers predated 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 / log(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 1, for all sufficiently large x. Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to prove
that there exists a prime number between n and 2n for any integer n ≥ 2.
An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "", the only paper he ever wrote on the subject. Riemann introduced new 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
of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of
to the study of the real function π(x) originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by
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.
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
(1949). While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, later proofs introduced various simplifications through the use of
but remained difficult to digest. A short proof was discovered in 1980 by American mathematician . Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses
from complex analysis.
Here is a sketch of the proof referred to in one of 's lectures. Like most proofs of the PNT, it starts out by reformulating the problem in terms of a less intuitive, but better-behaved, prime-counting function. The idea is to count the primes (or a related set such as the set of prime powers) with weights to arrive at a function with smoother asymptotic behavior. The most common such generalized counting function is the
ψ(x), defined by
 is prime
{\displaystyle \psi (x)=\!\!\!\!\sum _{\stackrel {p^{k}\leq x,}{p{\text{ is prime}}}}\!\!\!\!\log p.}
This is sometimes written as
{\displaystyle \psi (x)=\sum _{n\leq x}\Lambda (n),}
where Λ(n) is the , namely
 for some prime 
 and integer 
otherwise.
{\displaystyle \Lambda (n)={\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1,\\0&{\text{otherwise.}}\end{cases}}}
It is now relatively easy to check that the PNT is equivalent to the claim that
{\displaystyle \lim _{x\to \infty }{\frac {\psi (x)}{x}}=1.}
Indeed, this follows from the easy estimates
{\displaystyle \psi (x)=\sum _{p\leq x}\log p\left\lfloor {\frac {\log x}{\log p}}\right\rfloor \leq \sum _{p\leq x}\log x=\pi (x)\log x}
and (using ) for any ε & 0,
{\displaystyle \psi (x)\geq \!\!\!\!\sum _{x^{1-\varepsilon }\leq p\leq x}\!\!\!\!\log p\geq \!\!\!\!\sum _{x^{1-\varepsilon }\leq p\leq x}\!\!\!\!(1-\varepsilon )\log x=(1-\varepsilon )\left(\pi (x)+O\left(x^{1-\varepsilon }\right)\right)\log x.}
The next step is to find a useful representation for ψ(x). Let ζ(s) be the . It can be shown that ζ(s) is related to the von Mangoldt function Λ(n), and hence to ψ(x), via the relation
{\displaystyle -{\frac {\zeta '(s)}{\zeta (s)}}=\sum _{n=1}^{\infty }\Lambda (n)n^{-s}.}
A delicate analysis of this equation and related properties of the zeta function, using the
and , shows that for non-integer x the equation
{\displaystyle \psi (x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log(2\pi )}
holds, where the sum is over all zeros (trivial and nontrivial) of the zeta function. This striking formula is one of the so-called , and is already suggestive of the result we wish to prove, since the term x (claimed to be the correct asymptotic order of ψ(x)) appears on the right-hand side, followed by (presumably) lower-order asymptotic terms.
The next step in the proof involves a study of the zeros of the zeta function. The trivial zeros -2, -4, -6, -8, ... can be handled separately:
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2n\,x^{2n}}}=-{\frac {1}{2}}\log \left(1-{\frac {1}{x^{2}}}\right),}
which vanishes for a large x. The nontrivial zeros, namely those on the critical strip 0 ≤ Re(s) ≤ 1, can potentially be of an asymptotic order comparable to the main term x if Re(ρ) = 1, so we need to show that all zeros have real part strictly less than 1.
To do this, we take for granted that ζ(s) is meromorphic in the half-plane Re(s) & 0, and is analytic there except for a simple pole at s = 1, and that there is a product formula
{\displaystyle \zeta (s)=\prod _{p}{\frac {1}{1-p^{-s}}}}
for Re(s) & 1. This product formula follows from the existence of unique prime factorization of integers, and shows that ζ(s) is never zero in this region, so that its logarithm is defined there and
{\displaystyle \log \zeta (s)=-\sum _{p}\log \left(1-p^{-s}\right)=\sum _{p,n}{\frac {p^{-ns}}{n}}.}
Write s = x + iy; then
{\displaystyle {\big |}\zeta (x+iy){\big |}=\exp \left(\sum _{n,p}{\frac {\cos ny\log p}{np^{nx}}}\right).}
Now observe the identity
{\displaystyle 3+4\cos \phi +\cos 2\phi =2(1+\cos \phi )^{2}\geq 0,}
{\displaystyle \left|\zeta (x)^{3}\zeta (x+iy)^{4}\zeta (x+2iy)\right|=\exp \left(\sum _{n,p}{\frac {3+4\cos(ny\log p)+\cos(2ny\log p)}{np^{nx}}}\right)\geq 1}
for all x & 1. Suppose now that ζ(1 + iy) = 0. Certainly y is not zero, since ζ(s) has a simple pole at s = 1. Suppose that x & 1 and let x tend to 1 from above. Since
{\displaystyle \zeta (s)}
has a simple pole at s = 1 and ζ(x + 2iy) stays analytic, the left hand side in the previous inequality tends to 0, a contradiction.
Finally, we can conclude that the PNT is heuristically true. To rigorously complete the proof there are still serious technicalities to overcome, due to the fact that the summation over zeta zeros in the explicit formula for ψ(x) does not converge absolutely but only conditionally and in a "principal value" sense. There are several ways around this problem but many of them require rather delicate complex-analytic estimates that are beyond the scope of this paper. Edwards's book provides the details. Another method is to use , though this theorem is itself quite hard to prove. D. J. Newman observed that the full strength of Ikehara's theorem is not needed for the prime number theorem, and one can get away with a special case that is much easier to prove.
In a handwritten note on a reprint of his 1838 paper "Sur l'usage des séries infinies dans la théorie des nombres", which he mailed to ,
conjectured (under a slightly different form appealing to a series rather than an integral) that an even better approximation to π(x) is given by the
function Li(x), defined by
{\displaystyle \operatorname {Li} (x)=\int _{2}^{x}{\frac {dt}{\log t}}=\operatorname {li} (x)-\operatorname {li} (2).}
Indeed, this integral is strongly suggestive of the notion that the "density" of primes around t should be 1 / log t. This function is related to the logarithm by the
{\displaystyle \operatorname {Li} (x)\sim {\frac {x}{\log x}}\sum _{k=0}^{\infty }{\frac {k!}{(\log x)^{k}}}={\frac {x}{\log x}}+{\frac {x}{(\log x)^{2}}}+{\frac {2x}{(\log x)^{3}}}+\cdots }
So, the prime number theorem can also be written as π(x) ~ Li(x). In fact, in another paper in 1899 La Vallée Poussin proved that
{\displaystyle \pi (x)=\operatorname {Li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }
for some positive constant a, where O(...) is the . This has been improved to
{\displaystyle \pi (x)=\operatorname {Li} (x)+O\left(x\exp \left(-{\frac {A(\log x)^{\frac {3}{5}}}{(\log \log x)^{\frac {1}{5}}}}\right)\right).}
Because of the connection between the
and π(x), the
has considerable importance in : if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today. More specifically,
showed in 1901 that,
the Riemann hypothesis is true, the error term in the above relation can be improved to
{\displaystyle \pi (x)=\operatorname {Li} (x)+O\left({\sqrt {x}}\log x\right).}
The constant involved in the big O notation was estimated in 1976 by : assuming the Riemann hypothesis,
{\displaystyle {\big |}\pi (x)-\operatorname {li} (x){\big |}&{\frac {{\sqrt {x}}\log x}{8\pi }}}
for all x ≥ 2657. He also derived a similar bound for the
{\displaystyle {\big |}\psi (x)-x{\big |}&{\frac {{\sqrt {x}}(\log x)^{2}}{8\pi }}}
for all x ≥ 73.2. This latter bound has been shown to express a variance to mean
(when regarded as a random function over the integers),
and to also correspond to the . Parenthetically, the Tweedie distributions represent a family of
distributions that serve as foci of convergence for a generalization of the .
The logarithmic integral li(x) is larger than π(x) for "small" values of x. This is because it is (in some sense) counting not primes, but prime powers, where a power pn of a prime p is counted as 1/n of a prime. This suggests that li(x) should usually be larger than π(x) by roughly li(√x) / 2, and in particular should always be larger than π(x). However, in 1914,
proved that this is not the case. The first value of x where π(x) exceeds li(x) is probably around x = 10316; see the article on
for more details. (On the other hand, the
Li(x) is smaller than π(x) already for x = 2; indeed, Li(2) = 0, while π(2) = 1.)
In the first half of the twentieth century, some mathematicians (notably ) believed that there exists a hierarchy of proof methods in mathematics depending on what sorts of numbers (, , ) a proof requires, and that the prime number theorem (PNT) is a "deep" theorem by virtue of requiring . This belief was somewhat shaken by a proof of the PNT based on , though this could be set aside if Wiener's theorem were deemed to have a "depth" equivalent to that of complex variable methods.
In March 1948,
established, by "elementary" means, the asymptotic formula
{\displaystyle \vartheta (x)\log(x)+\sum \limits _{p\leq x}{\log(p)}\ \vartheta \left({\frac {x}{p}}\right)=2x\log(x)+O(x)}
{\displaystyle \vartheta (x)=\sum \limits _{p\leq x}{\log(p)}}
for primes p. By July of that year, Selberg and
had each obtained elementary proofs of the PNT, both using Selberg's asymptotic formula as a starting point. These proofs effectively laid to rest the notion that the PNT was "deep", and showed that technically "elementary" methods were more powerful than had been believed to be the case. On the history of the elementary proofs of the PNT, including the Erd?s–Selberg , see an article by .
There is some debate about the significance of Erd?s and Selberg's result. There is no rigorous and widely accepted definition of the notion of
in number theory, so it is not clear exactly in what sense their proof is "elementary". Although it does not use complex analysis, it is in fact much more technical than the standard proof of PNT. One possible definition of an "elementary" proof is "one that can be carried out in first order ." There are number-theoretic statements (for example, the ) provable using
methods, but such theorems are rare to date. Erd?s and Selberg's proof can certainly be formalized in Peano arithmetic, and in 1994, Charalambos Cornaros and Costas Dimitracopoulos proved that their proof can be formalized in a very weak fragment of PA, namely IΔ0 + exp, However, this does not address the question of whether or not the standard proof of PNT can be formalized in PA.
In 2005, Avigad et al. employed the
to devise a computer-verified variant of the Erd?s–Selberg proof of the PNT. This was the first machine-verified proof of the PNT. Avigad chose to formalize the Erd?s–Selberg proof rather than an analytic one because while Isabelle's library at the time could implement the notions of limit, derivative, and , it had almost no theory of integration to speak of.:19
In 2009, John Harrison employed
to formalize a proof employing . By developing the necessary analytic machinery, including the , Harrison was able to formalize "a direct, modern and elegant proof instead of the more involved 'elementary' Erd?s–Selberg argument".
Let πn,a(x) denote the number of primes in the
a, a + n, a + 2n, a + 3n, ... less than x.
conjectured, and
proved, that, if a and n are , then
{\displaystyle \pi _{n,a}(x)\sim {\frac {1}{\varphi (n)}}\operatorname {Li} (x),}
where φ is . In other words, the primes are distributed evenly among the residue classes [a]
n with gcd(a, n) = 1. This is stronger than
(which only states that there is an infinity of primes in each class) and can be proved using similar methods used by Newman for his proof of the prime number theorem.
gives a good estimate for the distribution of primes in residue classes.
Although we have in particular
{\displaystyle \pi _{4,1}(x)\sim \pi _{4,3}(x),}
empirically the primes congruent to 3 are more numerous and are nearly always ahead in this "prime number race"; the first reversal occurs at x = 26861.:1–2 However Littlewood showed in 1914:2 that there are infinitely many sign changes for the function
{\displaystyle \pi _{4,1}(x)-\pi _{4,3}(x),}
so the lead in the race switches back and forth infinitely many times. The phenomenon that π4,3(x) is ahead most of the time is called . The prime number race generalizes to other moduli and is the subj Pál Turán asked whether it is always the case that π(x;a,c) and π(x;b,c) change places when a and b are coprime to c.
and Martin give a thorough exposition and survey.
The prime number theorem is an asymptotic result. It gives an
bound on π(x) as a direct consequence of the definition of the limit: for all ε & 0, there is an S such that for all x & S,
{\displaystyle (1-\varepsilon ){\frac {x}{\log x}}&\pi (x)&(1+\varepsilon ){\frac {x}{\log x}}.}
However, better bounds on π(x) are known, for instance 's
{\displaystyle {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}\right)&\pi (x)&{\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2.51}{(\log x)^{2}}}\right).}
The first inequality holds for all x ≥ 599 and the second one for x ≥ 355991.
A weaker but sometimes useful bound for x ≥ 55 is
{\displaystyle {\frac {x}{\log x+2}}&\pi (x)&{\frac {x}{\log x-4}}.}
In 's thesis there are stronger versions of this type of inequality that are valid for larger x. Later in 2010, Dusart proved:
{\displaystyle {\begin{aligned}{\frac {x}{\log x-1}}&&\pi (x)&&{\text{for }}x\geq 5393,{\text{ and}}\\\pi (x)&&{\frac {x}{\log x-1.1}}&&{\text{for }}x\geq 60184.\end{aligned}}}
The proof by de la Vallée-Poussin implies the following. For every ε & 0, there is an S such that for all x & S,
{\displaystyle {\frac {x}{\log x-(1-\varepsilon )}}&\pi (x)&{\frac {x}{\log x-(1+\varepsilon )}}.}
th prime number[]
As a consequence of the prime number theorem, one gets an
expression for the nth prime number, denoted by pn:
{\displaystyle p_{n}\sim n\log n.}
A better approximation is
{\displaystyle {\frac {p_{n}}{n}}=\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}-{\frac {(\log \log n)^{2}-6\log \log n+11}{2(\log n)^{2}}}+o\left({\frac {1}{(\log n)^{2}}}\right).}
Again considering the 2×1017th prime number 8512677386048191063, this gives an estimate of 8512681315554715386; the first 5 digits match and relative error is about 0.00005%.
states that
{\displaystyle p_{n}&n\log n.}
This can be improved by the following pair of bounds:
{\displaystyle \log n+\log \log n-1&{\frac {p_{n}}{n}}&\log n+\log \log n\quad {\text{for }}n\geq 6.}
, x / log x, and li(x)[]
The table compares exact values of π(x) to the two approximations x / log x and li(x). The last column, x / π(x), is the average
below x.
π(x) - x/log x
π(x)/x / log x
li(x) - π(x)
20758029.0
4118054813
169923159.0
37607912018
1416705193.0
346065536839
11992858452.0
3204941750802
102838308636.0
29844570422669
891604962452.0
279238341033925
7804289844393.0
2623557157654233
68883734693281.0
24739954287740860
612483070893536.0
21949555.0
234057667276344607
5481624169369960.0
99877775.0
2220819602560918840
49347193044659701.0
222744644.0
21127269486018731928
446579871578168707.0
597394254.0
201467286689315906290
4060704006019620994.0
1932355208.0
1925320391606803968923
37083513766578631309.0
7250186216.0
18435599767349200867866
339996354713708049069.0
17146907278.0
176846309399143769411680
3128516637843038351228.0
55160980939.0
The value for π(1024) was originally co it has since been verified unconditionally.
There is an analogue of the prime number theorem that describes the "distribution" the form it takes is strikingly similar to the case of the classical prime number theorem.
To state it precisely, let F = GF(q) be the finite field with q elements, for some fixed q, and let Nn be the number of
irreducible polynomials over F whose
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
{\displaystyle N_{n}\sim {\frac {q^{n}}{n}}.}
If we make the substitution x = qn, then the right hand side is just
{\displaystyle {\frac {x}{\log _{q}x}},}
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 a monic polynomial of degree n is selected randomly, then the probability of it being irreducible is about 1/n.
One can even prove an analogue of the Riemann hypothesis, namely that
{\displaystyle N_{n}={\frac {q^{n}}{n}}+O\left({\frac {q^{\frac {n}{2}}}{n}}\right).}
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
{\displaystyle q^{n}=\sum _{d\mid n}dN_{d},}
where the sum is over all
then yields
{\displaystyle N_{n}={\frac {1}{n}}\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)q^{d},}
where μ(k) is the . (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
of n can be no larger than n/2.
for information about generalizations of the theorem.
for a generalization to prime ideals in algebraic number fields.
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visualizing the Prime Number Theorem.
by Chris Caldwell, .
by Tomás Oliveira e Silva
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