Prime-counting function

This article uses technical mathematical notation for logarithms. All instances of

log(x)

without a subscript base should be interpreted as a natural logarithm, also commonly written as

ln(x)

or

log e (x)

.

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$ .^[1]^[2] It is denoted by $π (x)$ (unrelated to the number $π$ ).

The values of $π (n)$ for the first 60 positive integers

Growth rate

Of great interest in number theory is the growth rate of the prime-counting function.^[3]^[4] It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately

{\frac {x}{\log(x)}}

where

log

is the natural logarithm, in the sense that

\lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\log(x)}}=1.

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

\lim _{x\rightarrow \infty }{\frac {\pi (x)}{\operatorname {li} (x)}}=1

where

li

is the logarithmic integral function. The prime number theorem 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. 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).^[5]

More precise estimates

In 1899, de la Vallée Poussin proved that ^[6]

\pi (x)=\operatorname {li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty

for some positive constant

a

. Here,

O (...)

is the big

O

notation.

More precise estimates of $π (x)$ are now known. For example, in 2002, Kevin Ford proved that^[7]

\pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-0.2098(\log x)^{3/5}(\log \log x)^{-1/5}\right)\right).

Mossinghoff and Trudgian proved^[8] an explicit upper bound for the difference between $π (x)$ and $li(x)$ :

{\bigl |}\pi (x)-\operatorname {li} (x){\bigr |}\leq 0.2593{\frac {x}{(\log x)^{3/4}}}\exp \left(-{\sqrt {\frac {\log x}{6.315}}}\right)\quad {\text{for }}x\geq 229.

For values of $x$ that are not unreasonably large, $li(x)$ is greater than $π (x)$ . However, $π (x) - li(x)$ is known to change sign infinitely many times. For a discussion of this, see Skewes' number.

Exact form

For $x > 1$ let $π0(x) = π(x) − .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2$ when $x$ is a prime number, and $π 0 (x) = π (x)$ otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that $π 0 (x)$ is equal to^[9]

\pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho }),

where

\operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} \left(x^{1/n}\right),

μ (n)

is the Möbius function,

li(x)

is the logarithmic integral function,

ρ

indexes every zero of the Riemann zeta function, and

li(x ρ / n)

is not evaluated with a branch cut but instead considered as

Ei(ρ / n log x)

where

Ei(x)

is the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros

ρ

of the Riemann zeta function, then

π 0 (x)

may be approximated by^[10]

\pi _{0}(x)\approx \operatorname {R} (x)-\sum _{\rho }\operatorname {R} \left(x^{\rho }\right)-{\frac {1}{\log(x)}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\log(x)}}.

The Riemann hypothesis suggests that every such non-trivial zero lies along $Re(s) = 1 / 2$ .

Table of $π (x)$ , $x / log(x)$ , 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 prime gap below $x$ .

Zdroj:https://en.wikipedia.org?pojem=Prime-counting_function
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Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok.
Podrobnejšie informácie nájdete na stránke Podmienky použitia.

$x$	$π (x)$	$π (x) - x / log(x)$	$li(x) - π (x)$	$x / log(x)$ % error	$li(x)$ % error	$x / π (x)$
10	4	0	2	8.22%	42.606%	2.500
10²	25	3	5	14.06%	18.597%	4.000
10³	168	23	10	14.85%	5.561%	5.952
10⁴	1,229	143	17	12.37%	1.384%	8.137
10⁵	9,592	906	38	9.91%	0.393%	10.425
10⁶	78,498	6,116	130	8.11%	0.164%	12.739
10⁷	664,579	44,158	339	6.87%	0.051%	15.047
10⁸	5,761,455	332,774	754	5.94%	0.013%	17.357
10⁹	50,847,534	2,592,592	1,701	5.23%	3.34×10⁻³ %	19.667
10¹⁰	455,052,511	20,758,029	3,104	4.66%	6.82×10⁻⁴ %	21.975
10¹¹

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

Growth rate

More precise estimates

Exact form

Table of π(x), x/log(x), and li(x)

Table of $π (x)$ , $x / log(x)$ , and $li(x)$