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Prime Number Theorem


PrimePi

The prime number theorem gives an asymptotic form for the prime counting function pi(n), which counts the number of primes less than some integer n. Legendre (1808) suggested that for large n,

 pi(n)∼n/(lnn+B),
(1)

with B=-1.08366 (where B is sometimes called Legendre's constant), a formula which is correct in the leading term only,

 n/(lnn+B)sinn/(lnn)-Bn/((lnn)^2)+B^2n/((lnn)^3)+...
(2)

(Nagell 1951, p. 54; Wagon 1991, pp. 28-29; Havil 2003, p. 177).

In 1792, when only 15 years old, Gauss proposed that

 pi(n)∼n/(lnn).
(3)

Gauss later refined his estimate to

 pi(n)∼Li(n),
(4)

where

 Li(n)=int_2^n(dx)/(lnx)
(5)

is the logarithmic integral. Gauss did not publish this result, which he first mentioned in an 1849 letter to Encke. It was subsequently posthumously published in 1863 (Gauss 1863; Havil 2003, pp. 174-176).

Note that Li(n) has the asymptotic series about infty of

Li(n)∼sum_(k=0)^(infty)(k!n)/((lnn)^(k+1))
(6)
∼n/(lnn)+n/((lnn)^2)+(2n)/((lnn)^3)+...,
(7)

and taking the first three terms has been shown to be a better estimate than n/lnn alone (Derbyshire 2004, pp. 116-117).

The statement (4) is often known as "the" prime number theorem and was proved independently by Hadamard (1896) and de la Vallée Poussin (1896). A plot of pi(n) (lower curve) and Li(n) is shown above for n<=1000.

For small n, it had been checked and always found that pi(n)<Li(n). As a result, many prominent mathematicians, including no less than both Gauss and Riemann, conjectured that the inequality was strict. To everyone's surprise, this conjecture was refuted when Littlewood (1914) proved that the inequality reverses infinitely often for sufficiently large n (Ball and Coxeter 1987; Havil 2003, p. 199). Skewes then showed that the first crossing of pi(n)-Li(n)=0 occurs before 10^(10^(10^(34))), a number now known as the Skewes number (Havil 2003, p. 199). The upper bound for the crossing has subsequently been reduced to 10^(371). Lehman (1966) proved that at least 10^(500) reversals occur for numbers with 1166 or 1167 decimal digits.

Chebyshev put limits on the ratio

 7/8<(pi(n))/(n/(lnn))<9/8
(8)

(Landau 1927; Nagell 1951, p. 55; Landau 1974; Hardy and Wright 1979, Ch. 22; Ingham 1990; Rubinstein and Sarnak 1994; Hardy 1999, p. 27; Derbyshire 2004, pp. 124 and 154). For large n, he showed that

 0.89Li(n)<pi(n)<1.11Li(n),
(9)

where Li(x) is the logarithmic integral (Edwards 2001, p. 4), and

 0.922<(pi(n))/(n/(lnn))<1.105
(10)

(Havil 2003, p. 186). He also showed that if the limit

 lim_(n->infty)(pi(n))/(n/(lnn))
(11)

exists, then it is 1 (Havil 2003, p. 186). Derbyshire's (2004, p. 124) statement that in 1850, Chebyshev also showed that pi(n) cannot differ from n/lnn by more than approximately 10% is therefore correct only for sufficiently large n.

Hadamard and de la Vallée Poussin independently proved the prime number theorem in 1896 by showing that the Riemann zeta function zeta(z) has no zeros of the form 1+it, in the sense that no deeper properties of zeta(s) are required for the proof (Smith 1994, p. 128; Hardy 1999, pp. 58-60). Wiener (1951) allowed this somewhat vague statement to be interpreted literally (Hardy 1999, pp. 34 and 46), and this proof was simplified by Landau (1932) and Bochner (1933).

An elementary proof was found by Erdős (1949) and Selberg (1950) (Ball and Coxeter 1987, p. 63; Havil 2003, p. 188), although an unfortunate priority dispute over the joint work marred the otherwise beautiful proof (Hoffman 1998, pp. 39-41; Derbyshire 2004, p. 125). Versions of elementary proofs of the prime number theorem appear in final section of Nagell (1951) and in Hardy and Wright (1979, pp. 359-367). As noted by Hardy and Wright (1979, p. 9), although this proof is 'elementary,' "this proof is not easy."

Hadamard's proof depends on the simple trigonometric inequality

 3+4costheta+cos(2theta)=2(1+costheta)^2>=0
(12)

(Hardy 1999, p. 58; Havil 2003, p. 187). de la Vallée Poussin (1899) showed that

 pi(x)=Li(x)+O(x/(lnx)e^(-asqrt(lnx)))
(13)

for some constant a (Knuth 1998, p. 381), where O(x) is asymptotic notation. In 1901, Koch showed that if the Riemann hypothesis is true, then

 pi(x)=Li(x)+O(sqrt(x)lnx)
(14)

(Havil 2003, p. 205), which can be written in the slightly weaker form

 pi(x)=Li(x)+O(x^(1/2+epsilon))
(15)

(Derbyshire 2004, pp. 237 and 242-244).

The error term in (15) has subsequently been improved to

 pi(x)=Li(x)+O(xexp(-(A(lnx)^(3/5))/((lnlnx)^(1/5))))
(16)

(Walfisz 1963; Riesel 1994, p. 56; Knuth 1998, p. 382; Derbyshire 2004, p. 244). Ingham (1930) proved the prime number theorem using the identity of Ramanujan

 sum_(n=1)^infty(sigma_a(n)sigma_b(n))/(n^s)=(zeta(s)zeta(s-a)zeta(s-b)zeta(s-a-b))/(zeta(2s-a-b)),
(17)

where sigma_a(n) is the divisor function (Hardy 1999, pp. 59-60).

Riemann estimated the prime counting function with

 pi(n)∼Li(n)-1/2Li(n^(1/2)),
(18)

which is a better approximation than Li(n) for n<10^7. Riemann (1859) also suggested the Riemann function

 R(x)=sum_(n=1)^infty(mu(n))/nLi(x^(1/n)),
(19)

where mu is the Möbius function (Wagon 1991, p. 29). An even better approximation for small n (by a factor of 10 for n<10^9) is the Gram series.

The prime number theorem is equivalent to either

 lim_(x->infty)(theta(x))/x=1
(20)

or

 lim_(x->infty)(psi(x))/x=1,
(21)

where theta(x) and psi(x) are the Chebyshev functions. Chebyshev showed that the only possible limit of these expressions was 1, but was not able to prove existence of the limit (Hardy 1999, p. 28).

The Riemann hypothesis is equivalent to the assertion that

 |Li(x)-pi(x)|<=csqrt(x)lnx
(22)

for some value of c (Ingham 1990, p. 83; Landau 1974, pp. 378-388; Ball and Coxeter 1987; Hardy 1999, p. 26), as shown by Koch in 1901 (Havil 2003, p. 205). Some limits obtained without assuming the Riemann hypothesis are

pi(x)=Li(x)+O[xe^(-(lnx)^(1/2)/15)]
(23)
pi(x)=Li(x)+O[xe^(-0.009(lnx)^(3/5)/(lnlnx)^(1/5))].
(24)

See also

Bertrand's Postulate, Chebyshev Functions, Chebyshev's Theorem, Dirichlet's Theorem, Gram Series, Prime Counting Function, Riemann Function, Selberg's Formula, Skewes Number Explore this topic in the MathWorld classroom

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References

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Cite this as:

Weisstein, Eric W. "Prime Number Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimeNumberTheorem.html

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