Prime-counting function

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by $\scriptstyle \pi (x)$ (this does not refer to the number π).

The 60 first values of π(n)

History

Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately

$x/\operatorname {ln} (x)\!$

in the sense that

$\lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\operatorname {ln} (x)}}=1.\!$

This statement is the prime number theorem. An equivalent statement is

$\lim _{x\rightarrow \infty }\pi (x)/\operatorname {li} (x)=1\!$

where li is the logarithmic integral function. This was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859.

More precise estimates of $\pi (x)\!$ are now known; for example

$\pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-{\frac {\sqrt {\ln(x)}}{15}}\right)\right)\!$

where the O is big O notation. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).

Another conjecture about the growth rate for prime series involving the prime number theorem is

Failed to parse (syntax error): {\displaystyle \sum_{p \le x} p^{n} \sim \pi(x^{n+1})=Li(x^{n+1}).\}

Table of π(x), x / ln x, and Li(x)

The table shows how the three functions π(x), x / ln x and Li(x) compare at powers of 10.

x	π(x)	π(x) − x / ln x	Li(x) − π(x)	x / π(x)
10	4	−0.3	2.2	2.500
102	25	3.3	5.1	4.000
103	168	23	10	5.952
104	1,229	143	17	8.137
105	9,592	906	38	10.425
106	78,498	6,116	130	12.740
107	664,579	44,158	339	15.047
108	5,761,455	332,774	754	17.357
109	50,847,534	2,592,592	1,701	19.667
1010	455,052,511	20,758,029	3,104	21.975
1011	4,118,054,813	169,923,159	11,588	24.283
1012	37,607,912,018	1,416,705,193	38,263	26.590
1013	346,065,536,839	11,992,858,452	108,971	28.896
1014	3,204,941,750,802	102,838,308,636	314,890	31.202
1015	29,844,570,422,669	891,604,962,452	1,052,619	33.507
1016	279,238,341,033,925	7,804,289,844,393	3,214,632	35.812
1017	2,623,557,157,654,233	68,883,734,693,281	7,956,589	38.116
1018	24,739,954,287,740,860	612,483,070,893,536	21,949,555	40.420
1019	234,057,667,276,344,607	5,481,624,169,369,960	99,877,775	42.725
1020	2,220,819,602,560,918,840	49,347,193,044,659,701	222,744,644	45.028
1021	21,127,269,486,018,731,928	446,579,871,578,168,707	597,394,254	47.332
1022	201,467,286,689,315,906,290	4,060,704,006,019,620,994	1,932,355,208	49.636
1023	1,925,320,391,606,803,968,923	37,083,513,766,578,631,309	7,250,186,216	51.939

The π(x) column is sequence A006880 in OEIS; x / ln x is sequence A057835; and Li(x) is sequence A057752. The value for π(10²³) is by T. O. e Silva.

Algorithms for evaluating π(x)

A simple way to find $\pi (x)$, if $x$ is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to $x$ and then to count them.

A more elaborate way of finding $\pi (x)$ is due to Legendre: given $x$, if $p_{1}$, $p_{2}$, …, $p_{k}$ are distinct prime numbers, then the number of integers less than or equal to $x$ which are divisible by no $p_{i}$ is

$\lfloor x\rfloor -\sum _{i}\left\lfloor {\frac {x}{p_{i}}}\right\rfloor +\sum _{i<j}\left\lfloor {\frac {x}{p_{i}p_{j}}}\right\rfloor -\sum _{i<j<k}\left\lfloor {\frac {x}{p_{i}p_{j}p_{k}}}\right\rfloor +\cdots ,$

(where $\lfloor \cdot \rfloor$ denotes the floor function). This number is therefore equal to

$\pi (x)-\pi \left({\sqrt {x}}\right)+1\,$

when the numbers $p_{1},p_{2},\dots ,p_{k}$ are the prime numbers less than or equal to the square root of $x$.

In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating $\pi (x)$. Let $p_{1}$, $p_{2}$, …, $p_{n}$ be the first $n$ primes and denote by $\Phi (m,n)$ the number of natural numbers not greater than $m$ which are divisible by no $p_{i}$. Then

$\Phi (m,n)=\Phi (m,n-1)-\Phi \left(\left[{\frac {m}{p_{n}}}\right],n-1\right).\,$

Given a natural number $m$, if $n=\pi \left({\sqrt[{3}]{m}}\right)$ and if $\mu =\pi \left({\sqrt {m}}\right)-n$, then

$\pi (m)=\Phi (m,n)+n(\mu +1)+{\frac {\mu ^{2}-\mu }{2}}-1-\sum _{k=1}^{\mu }\pi \left({\frac {m}{p_{n+k}}}\right).\,$

Using this approach, Meissel computed $\pi (x)$, for $x$ equal to 5×10⁵, 10⁶, 10⁷, and 10⁸.

In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real $m$ and for natural numbers $n$, and $k$, $P_{k}(m,n)$ as the number of numbers not greater than m with exactly k prime factors, all greater than $p_{n}$. Furthermore, set $P_{0}(m,n)=1$. Then

$\Phi (m,n)=\sum _{k=0}^{+\infty }P_{k}(m,n),\,$

where the sum actually has only finitely many nonzero terms. Let $y$ denote an integer such that ${\sqrt[{3}]{m}}\leq y\leq {\sqrt {m}}$, and set $n=\pi (y)$. Then $P_{1}(m,n)=\pi (m)-n$ and $P_{k}(m,n)=0$ when $k$ ≥ 3. Therefore

$\pi (m)=\Phi (m,n)+n-1-P_{2}(m,n).$

The computation of $P_{2}(m,n)$ can be obtained this way:

$P_{2}(m,n)=\sum _{y<p\leq {\sqrt {m}}}\left(\pi \left({\frac {m}{p}}\right)-\pi (p)+1\right).\,$

On the other hand, the computation of $\Phi (m,n)$ can be done using the following rules:

1. $\Phi (m,0)=\lfloor m\rfloor ;\,$
2. $\Phi (m,b)=\Phi (m,b-1)-\Phi \left({\frac {m}{p_{b}}},b-1\right).\,$

Using his method and an IBM 701, Lehmer was able to compute $\pi \left(10^{10}\right)$.

The Chinese mathematician Hwang Cheng, in a conference about prime number functions at the University of Bordeaux, used the following identities:

$e^{(a-1)\Theta }f(x)=f(ax),\,$

$J(x)=\sum _{n=1}^{\infty }{\frac {\pi (x^{1/n})}{n}},$

and setting $x=e^{t}$, Laplace-transforming both sides and applying a geometric sum on $e^{n\Theta }$ got the expression

${\frac {1}{2{\pi }i}}\int _{c-i\infty }^{c+i\infty }g(s)t^{s}\,ds=\pi (t),$

${\frac {\ln \zeta (s)}{s}}=(1-e^{\Theta (s)})^{-1}g(s)$

$\Theta (s)=s{\frac {d}{ds}}.$

Other prime-counting functions

Other prime-counting functions are also used because they are more convenient to work with. One is Riemann's prime-counting function, usually denoted as $\Pi _{0}(x)$. This has jumps of 1/n for prime powers pⁿ, with it taking a value half-way between the two sides at discontinuities. That added detail is because then it may be defined by an inverse Mellin transform. Formally, we may define $\Pi _{0}(x)$ by

$\Pi _{0}(x)={\frac {1}{2}}{\bigg (}\sum _{p^{n}<x}{\frac {1}{n}}\ +\sum _{p^{n}\leq x}{\frac {1}{n}}{\bigg )}$

where p is a prime.

We may also write

$\Pi _{0}(x)=\sum _{2}^{x}{\frac {\Lambda (n)}{\ln n}}-{\frac {1}{2}}{\frac {\Lambda (x)}{\ln x}}=\sum _{n=1}^{\infty }{\frac {1}{n}}\pi _{0}(x^{1/n})$

where Λ(n) is the von Mangoldt function and

$\pi _{0}(x)={\frac {\pi (x-0)+\pi (x+0)}{2}}.$

Möbius inversion formula then gives

$\pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi _{0}(x^{1/n})=\int _{1}^{\infty }duM'(u)\Pi _{0}(x^{1/u})u^{-1}$

where M(u) is the Mertens function.

Knowing the relationship between log of the Riemann zeta function and the von Mangoldt function $\Lambda$, and using the Perron formula we have

$\ln \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)x^{-s+1}\,dx$

The Chebyshev function weights primes or prime powers pⁿ by ln(p):

$\theta (x)=\sum _{p\leq x}\ln p$

$\psi (x)=\sum _{p^{n}\leq x}\ln p=\sum _{n=1}^{\infty }\theta (x^{1/n})=\sum _{n\leq x}\Lambda (n).$

Formulas for prime-counting functions

These come in two kinds, arithmetic formulas and analytic formulas. The latter are what allow us to prove the prime number theorem. They stem from the work of Riemann and von Mangoldt, and are generally known as explicit formulas.

We have the following expression for ψ:

$\psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln 2\pi -{\frac {1}{2}}\ln(1-x^{-2})$

where

$\psi _{0}(x)=\lim _{\epsilon \rightarrow 0}{\frac {\psi (x-\epsilon )+\psi (x+\epsilon )}{2}}.$

Here ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero and one. The formula is valid for values of x greater than one, which is the region of interest. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last subtrahend in the formula.

For $\scriptstyle \Pi _{0}(x)$ we have a more complicated formula

$\Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} (x^{\rho })-\ln 2+\int _{x}^{\infty }{\frac {dt}{t(t^{2}-1)\ln t}}.$

Again, the formula is valid for x > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value, and, again, the latter integral, taken with minus sign, is just the same sum, but over the trivial zeros. The first term li(x) is the usual logarithmic integral function; the expression li(x^ρ) in the second term should be considered as Ei(ρ ln x), where Ei is the analytic continuation of the exponential integral function from positive reals to the complex plane with branch cut along the negative reals.

Thus, Möbius inversion formula gives us

$\pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-{\frac {1}{\ln x}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}$

valid for x > 1, where

$\operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n})=1+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k!k\zeta (k+1)}}$

is so-called Riemann's R-function. The latter series for it is known as Gram series and converges for all positive x.

Delta function (red line) on log scale

The sum over non-trivial zeta zeros in the formula for $\scriptstyle \pi _{0}(x)$ describes the fluctuations of $\scriptstyle \pi _{0}(x)$, while remaining terms give the "smooth" part of prime-counting function. The amplitude of the "noisy" part is heuristically about $\scriptstyle {\sqrt {x}}/\ln x$, so the fluctuations of the distribution of primes can be represented with the Delta function:

$\Delta (x)=\left(\pi _{0}(x)-\operatorname {R} (x)+{\frac {1}{\ln x}}-{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}\right){\frac {\ln x}{\sqrt {x}}}.$

An extensive table of the values of Δ(x), based on the results of Tomás Oliveira e Silva, is available.

Inequalities

Here are some useful inequalities for π(x).

$\pi (x)<1.25506{\frac {x}{\log x}}\!$ for x > 1.

${\frac {x}{\log x+2}}<\pi (x)<{\frac {x}{\log x-4}}\!$ for x ≥ 55.

Here are some inequalities for the n^th prime, p_n.

$n\ \ln n+n\ln \ln n-n<p_{n}<n\ln n+n\ln \ln n\!$ for n ≥ 6.

The left inequality holds for n ≥ 1 and the right inequality holds for n ≥ 6.

An approximation for the n^th prime number is

$p_{n}=n\ln n+n\ln \ln n-n+{\frac {n\ln \ln n-2n}{\ln n}}+O\left({\frac {n(\ln \ln n)^{2}}{(\ln n)^{2}}}\right).$

The Riemann hypothesis

The Riemann hypothesis is equivalent to a much tighter bound on the error in the estimate for $\pi (x)$, and hence to a more regular distribution of prime numbers,

$\pi (x)=\operatorname {li} (x)+O({\sqrt {x}}\log {x}).$

Relation to prime sums

if we had a sum of a function over all primes

$\sum _{p}^{}f(x)\!$

and we wish to accelerate its convergence we can write it as:

$\sum _{n=1}^{\infty }(-1)^{n}(\pi (n)-\pi (n-1)+1)f(n)=2f(2)-\sum _{p}f(x)+\sum _{n=1}^{\infty }(-1)^{n}f(n)\!$

for the series on the left we could apply Euler transform for alternating series, providing that f(n)>f(n+1) and that the 2 series converges, it also relates an alternating series to its prime sum counterpart, the main task of using this is that we can give a good approximation using only a few values of the prime number counting function.

References

• Bach, Eric (1996). Algorithmic Number Theory. MIT Press. pp. volume 1 page 234 section 8.8. ISBN 0-262-02405-5. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Marc Deléglise and Jöel Rivat, Computing $\pi (x)$: The Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Mathematics of Computation, vol. 65, number 33, January 1996, pages 235–245

• Dickson, Leonard Eugene (2005). History of the Theory of Numbers I: Divisibility and Primality. Dover Publications. ISBN 0-486-44232-2.

• Ireland, Kenneth (1998). A Classical Introduction to Modern Number Theory (Second edition ed.). Springer. ISBN 0-387-97329-X. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Weisstein, Eric W. "Riemann Prime Counting Function". MathWorld.

• Hwang H. Cheng Prime Magic conference given at the University of Bordeaux (France) at year 2001 Démarches de la Géométrie et des Nombres de l'Université du Bordeaux

• Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, 1960.

• Oliveira e Silva, Tomás Tables of values of pi(x) and of pi2(x)

• Gourdon, Xavier; Sebah,Pascal PrimePi values thru 4E22

External Links

• Chris Caldwell, The Nth Prime Page at The Prime Pages.