prime-counting function

{{main|Prime number theorem}}

Of great interest in number theory is the growth rate of the prime-counting function.{{cite web | publisher=Chris K. Caldwell | title=How many primes are there? | url=http://primes.utm.edu/howmany.shtml | access-date=2008-12-02 | archive-date=2012-10-15 | archive-url=https://web.archive.org/web/20121015002415/http://primes.utm.edu/howmany.shtml | url-status=dead }}{{cite book |author-link=L. E. Dickson| first=Leonard Eugene | last=Dickson | year=2005 | title=History of the Theory of Numbers, Vol. I: Divisibility and Primality | publisher=Dover Publications | isbn=0-486-44232-2}} It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately

$$\backslash frac\{x\}\{\backslash log\; x\}$$

where {{math|log}} is the natural logarithm, in the sense that

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

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

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

where {{math|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).{{cite book | first=Kenneth | last=Ireland |author2=Rosen, Michael | year=1998 | title=A Classical Introduction to Modern Number Theory | edition=Second | publisher=Springer | isbn=0-387-97329-X }}

In 1899, de la Vallée Poussin proved that

See also Theorem 23 of {{cite book |author = A. E. Ingham |author-link = Albert Ingham |title = The Distribution of Prime Numbers |date=2000 |publisher = Cambridge University Press |isbn=0-521-39789-8}}

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

for some positive constant {{mvar|a}}. Here, {{math|O(...)}} is the big O notation.

More precise estimates of {{math|π(x)}} are now known. For example, in 2002, Kevin Ford proved that{{cite journal |author = Kevin Ford |title=Vinogradov's Integral and Bounds for the Riemann Zeta Function |journal=Proc. London Math. Soc. |date=November 2002 |volume=85 |issue=3 |pages=565–633 |url=https://faculty.math.illinois.edu/~ford/wwwpapers/zetabd.pdf |doi=10.1112/S0024611502013655 |arxiv=1910.08209 |s2cid=121144007 }}

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

Mossinghoff and Trudgian proved{{cite journal | first1 = Michael J. | last1 = Mossinghoff | first2 = Timothy S. | last2 = Trudgian | author2-link=Timothy Trudgian| title = Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function | journal = J. Number Theory | volume = 157 | year = 2015 | pages = 329–349 | arxiv = 1410.3926 | doi = 10.1016/J.JNT.2015.05.010| s2cid = 117968965 }} an explicit upper bound for the difference between {{math|π(x)}} and {{math|li(x)}}:

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

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

For {{math|x > 1}} let {{math|π₀(x) {{=}} π(x) − {{sfrac|1|2}}}} when {{mvar|x}} is a prime number, and {{math|π₀(x) {{=}} π(x)}} otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that {{math|π₀(x)}} is equal to{{Cite web|url=http://ism.uqam.ca/~ism/pdf/Hutama-scientific%20report.pdf|title=Implementation of Riemann's Explicit Formula for Rational and Gaussian Primes in Sage|last=Hutama|first=Daniel|date=2017|website=Institut des sciences mathématiques}}File:Riemann Explicit Formula.gif

$$\backslash pi\_0(x)\; =\; \backslash operatorname\{R\}(x)\; -\; \backslash sum\_\{\backslash rho\}\backslash operatorname\{R\}(x^\backslash rho),$$

where

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

{{math|μ(n)}} is the Möbius function, {{math|li(x)}} is the logarithmic integral function, {{mvar|ρ}} indexes every zero of the Riemann zeta function, and {{math|li(x^{{{sfrac|ρ|n}}})}} is not evaluated with a branch cut but instead considered as {{math|Ei({{sfrac|ρ|n}} log x)}} where {{math|Ei(x)}} is the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros {{mvar|ρ}} of the Riemann zeta function, then {{math|π₀(x)}} may be approximated by{{Cite journal | author1-link=Hans Riesel | last1=Riesel | first1=Hans | last2=Göhl | first2=Gunnar | title=Some calculations related to Riemann's prime number formula | doi=10.2307/2004630 | mr=0277489 | year=1970 | journal=Mathematics of Computation | issn=0025-5718 | volume=24 | issue=112 | pages=969–983 | jstor=2004630 | publisher=American Mathematical Society |url=https://www.ams.org/journals/mcom/1970-24-112/S0025-5718-1970-0277489-3/S0025-5718-1970-0277489-3.pdf }}

$$\backslash pi\_0(x)\; \backslash approx\; \backslash operatorname\{R\}(x)\; -\; \backslash sum\_\{\backslash rho\}\backslash operatorname\{R\}\backslash left(x^\backslash rho\backslash right)\; -\; \backslash frac\{1\}\{\backslash log\; x\}\; +\; \backslash frac\{1\}\{\backslash pi\}\; \backslash arctan\{\backslash frac\{\backslash pi\}\{\backslash log\; x\}\}\; .$$

The Riemann hypothesis suggests that every such non-trivial zero lies along {{math|1=Re(s) = {{sfrac|1|2}}}}.

The table shows how the three functions {{math|π(x)}}, {{math|{{sfrac|x|log x}}}}, and {{math|li(x)}} compared at powers of 10. See also,{{cite web |title=Tables of values of {{math|π(x)}} and of {{math|π₂(x)}} |url=https://sweet.ua.pt/tos/primes.html |publisher=Tomás Oliveira e Silva |access-date=2024-03-31}} and{{cite web |title=A table of values of {{math|π(x)}} |url=http://numbers.computation.free.fr/Constants/Primes/pixtable.html |publisher=Xavier Gourdon, Pascal Sebah, Patrick Demichel |access-date=2008-09-14}}

:

File:Prime number theorem ratio convergence.svg

In the On-Line Encyclopedia of Integer Sequences, the {{math|π(x)}} column is sequence {{OEIS2C|id=A006880}}, {{math| π(x) − {{sfrac|x|log x}}}} is sequence {{OEIS2C|id=A057835}}, and {{math|li(x) − π(x)}} is sequence {{OEIS2C|id=A057752}}.

The value for {{math|π(10²⁴)}} was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.{{cite web |title=Conditional Calculation of π(10²⁴) |first=Jens |last=Franke |author-link=Jens Franke |date=2010-07-29 |url=https://t5k.org/notes/pi(10to24).html |publisher=Chris K. Caldwell |access-date=2024-03-30}}

It was later verified unconditionally in a computation by D. J. Platt.{{cite journal |title=Computing {{math|π(x)}} Analytically |arxiv=1203.5712 |last1=Platt |first1=David J. |journal=Mathematics of Computation |volume=84 |issue=293 |date=May 2015 |orig-date=March 2012 |pages=1521–1535 |doi=10.1090/S0025-5718-2014-02884-6 |doi-access=free}}

The value for {{math|π(10²⁵)}} is by the same four authors.{{cite web |title=Analytic Computation of the prime-counting Function |url=http://www.math.uni-bonn.de/people/jbuethe/topics/AnalyticPiX.html |publisher=J. Buethe |date=27 May 2014 |access-date=2015-09-01}} Includes 600,000 value of {{math|π(x)}} for {{math|10¹⁴ ≤ x ≤ 1.6×10¹⁸}}

The value for {{math|π(10²⁶)}} was computed by D. B. Staple.{{cite thesis |title=The combinatorial algorithm for computing π(x) |date=19 August 2015 |url=http://dalspace.library.dal.ca/handle/10222/60524 |publisher=Dalhousie University |access-date=2015-09-01|type=Thesis |last1=Staple |first1=Douglas }} All other prior entries in this table were also verified as part of that work.

The values for 10²⁷, 10²⁸, and 10²⁹ were announced by David Baugh and Kim Walisch in 2015,{{cite web|website=Mersenne Forum|first=Kim |last=Walisch|title=New confirmed π(10²⁷) prime counting function record |date=September 6, 2015|url=http://www.mersenneforum.org/showthread.php?t=20473}} 2020,{{cite web |last=Baugh |first=David |date=August 30, 2020 |title=New prime counting function record, pi(10^28) |url=https://www.mersenneforum.org/showpost.php?p=555434&postcount=28 |website=Mersenne Forum}} and 2022,{{cite web |first=Kim |last=Walisch |date=March 4, 2022 |title=New prime counting function record: PrimePi(10^29) |url=https://www.mersenneforum.org/showpost.php?p=601061&postcount=38 |website=Mersenne Forum}} respectively.

A simple way to find {{math|π(x)}}, if {{mvar|x}} is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to {{mvar|x}} and then to count them.

A more elaborate way of finding {{math|π(x)}} is due to Legendre (using the inclusion–exclusion principle): given {{mvar|x}}, if {{math|p₁, p₂,…, p_n}} are distinct prime numbers, then the number of integers less than or equal to {{mvar|x}} which are divisible by no {{mvar|p_i}} is

:

(where {{math|⌊x⌋}} denotes the floor function). This number is therefore equal to

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

when the numbers {{math|p₁, p₂,…, p_n}} are the prime numbers less than or equal to the square root of {{mvar|x}}.

{{main|Meissel–Lehmer algorithm}}

In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating {{math|π(x)}}: Let {{math|p₁, p₂,…, p_n}} be the first {{mvar|n}} primes and denote by {{math|Φ(m,n)}} the number of natural numbers not greater than {{mvar|m}} which are divisible by none of the {{mvar|p_i}} for any {{math|i ≤ n}}. Then

: $\backslash Phi(m,n)=\backslash Phi(m,n-1)-\backslash Phi\backslash left(\backslash frac\; m\; \{p\_n\},n-1\backslash right).$

Given a natural number {{mvar|m}}, if {{math|n {{=}} π({{sqrt|m|3}})}} and if {{math|μ {{=}} π({{sqrt|m}}) − n}}, then

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

Using this approach, Meissel computed {{math|π(x)}}, for {{mvar|x}} equal to {{val|5e5}}, 10⁶, 10⁷, and 10⁸.

In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real {{mvar|m}} and for natural numbers {{mvar|n}} and {{mvar|k}}, {{math|P_k(m,n)}} as the number of numbers not greater than {{mvar|m}} with exactly {{mvar|k}} prime factors, all greater than {{mvar|p_n}}. Furthermore, set {{math|P₀(m,n) {{=}} 1}}. Then

:$\backslash Phi(m,n)\; =\; \backslash sum\_\{k=0\}^\{+\backslash infty\}\; P\_k(m,n)$

where the sum actually has only finitely many nonzero terms. Let {{mvar|y}} denote an integer such that {{math|{{sqrt|m|3}} ≤ y ≤ {{sqrt|m}}}}, and set {{math|n {{=}} π(y)}}. Then {{math|P₁(m,n) {{=}} π(m) − n}} and {{math|P_k(m,n) {{=}} 0}} when {{math|k ≥ 3}}. Therefore,

:$\backslash pi(m)\; =\; \backslash Phi(m,n)\; +\; n\; -\; 1\; -\; P\_2(m,n)$

The computation of {{math|P₂(m,n)}} can be obtained this way:

:

where the sum is over prime numbers.

On the other hand, the computation of {{math|Φ(m,n)}} can be done using the following rules:

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

Using his method and an IBM 701, Lehmer was able to compute the correct value of {{math|π(10⁹)}} and missed the correct value of {{math|π(10¹⁰)}} by 1.{{cite journal |last=Lehmer |first=Derrick Henry |author-link=D. H. Lehmer |date=1 April 1958 |title=On the exact number of primes less than a given limit |journal=Illinois J. Math. |volume=3 |issue=3 |pages=381–388 |url=https://projecteuclid.org/download/pdf_1/euclid.ijm/1255455259 |access-date=1 February 2017 }}

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.{{cite journal |author1 = Deléglise, Marc |author2 = Rivat, Joel |date=January 1996 |title=Computing {{math|π(x)}}: The Meissel, Lehmer, Lagarias, Miller, Odlyzko method |journal=Mathematics of Computation |volume=65 |issue=213 |pages=235–245 |doi = 10.1090/S0025-5718-96-00674-6 |doi-access=free |url=https://www.ams.org/mcom/1996-65-213/S0025-5718-96-00674-6/S0025-5718-96-00674-6.pdf }}

Other prime-counting functions are also used because they are more convenient to work with.

Riemann's prime-power counting function is usually denoted as {{math|Π₀(x)}} or {{math|J₀(x)}}. It has jumps of {{math|{{sfrac|1|n}}}} at prime powers {{mvar|pⁿ}} and it takes a value halfway between the two sides at the discontinuities of {{math|π(x)}}. That added detail is used because the function may then be defined by an inverse Mellin transform.

Formally, we may define {{math|Π₀(x)}} by

:

where the variable {{mvar|p}} in each sum ranges over all primes within the specified limits.

We may also write

:$\backslash \; \backslash Pi\_0(x)\; =\; \backslash sum\_\{n=2\}^x\; \backslash frac\{\backslash Lambda(n)\}\{\backslash log\; n\}\; -\; \backslash frac\{\backslash Lambda(x)\}\{2\backslash log\; x\}\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\; 1\; n\; \backslash pi\_0\backslash left(x^\{1/n\}\backslash right)$

where {{math|Λ}} is the von Mangoldt function and

:$\backslash pi\_0(x)\; =\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash frac\{\backslash pi(x-\backslash varepsilon)\; +\; \backslash pi(x+\backslash varepsilon)\}\{2\}.$

The Möbius inversion formula then gives

:$\backslash pi\_0(x)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash mu(n)\}\{n\}\backslash \; \backslash Pi\_0\backslash left(x^\{1/n\}\backslash right),$

where {{math|μ(n)}} is the Möbius function.

Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function {{math|Λ}}, and using the Perron formula we have

:$\backslash log\; \backslash zeta(s)\; =\; s\; \backslash int\_0^\backslash infty\; \backslash Pi\_0(x)\; x^\{-s-1\}\backslash ,\; \backslash mathrm\{d\}x$

The Chebyshev function weights primes or prime powers {{mvar|pⁿ}} by {{math|log p}}:

:$\backslash begin\{align\}$

\vartheta(x) &= \sum_{p\le x} \log p \\

\psi(x)&=\sum_{p^n \le x} \log p = \sum_{n=1}^\infty \vartheta \left( x^{1/n} \right) = \sum_{n \le x}\Lambda(n) .

\end{align}

For {{math|x ≥ 2}},{{cite book |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |year=2010 |title=Introduction to Analytic Number Theory |publisher=Springer |isbn= 978-1441928054}}

:$\backslash vartheta(x)\; =\; \backslash pi(x)\backslash log\; x\; -\; \backslash int\_2^x\; \backslash frac\{\backslash pi(t)\}\{t\}\backslash ,\; \backslash mathrm\{d\}t$

and

:$\backslash pi(x)=\backslash frac\{\backslash vartheta(x)\}\{\backslash log\; x\}\; +\; \backslash int\_2^x\; \backslash frac\{\backslash vartheta(t)\}\{t\backslash log^\{2\}(t)\}\; \backslash mathrm\{d\}\; t\; .$

Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting were the first used to prove the prime number theorem. They stem from the work of Riemann and von Mangoldt, and are generally known as explicit formulae.{{cite book |first=E.C. |last=Titchmarsh |year=1960 |title=The Theory of Functions, 2nd ed. |publisher=Oxford University Press}}

We have the following expression for the second Chebyshev function {{mvar|ψ}}:

:$\backslash psi\_0(x)\; =\; x\; -\; \backslash sum\_\backslash rho\; \backslash frac\{x^\backslash rho\}\{\backslash rho\}\; -\; \backslash log\; 2\backslash pi\; -\; \backslash frac\{1\}\{2\}\; \backslash log\backslash left(1-x^\{-2\}\backslash right),$

where

: $\backslash psi\_0(x)\; =\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0\}\; \backslash frac\{\backslash psi(x\; -\; \backslash varepsilon)\; +\; \backslash psi(x\; +\; \backslash varepsilon)\}\{2\}.$

Here {{mvar|ρ}} are the zeros of the Riemann zeta function in the critical strip, where the real part of {{mvar|ρ}} is between zero and one. The formula is valid for values of {{mvar|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 {{math|Π₀(x)}} we have a more complicated formula

:$\backslash Pi\_0(x)\; =\; \backslash operatorname\{li\}(x)\; -\; \backslash sum\_\backslash rho\; \backslash operatorname\{li\}\backslash left(x^\backslash rho\backslash right)\; -\; \backslash log\; 2\; +\; \backslash int\_x^\backslash infty\; \backslash frac\{\backslash mathrm\{d\}t\}\{t\; \backslash left(t^2\; -\; 1\backslash right)\; \backslash log\; t\}.$

Again, the formula is valid for {{math|x > 1}}, while {{mvar|ρ}} are the nontrivial zeros of the zeta function ordered according to their absolute value. The first term {{math|li(x)}} is the usual logarithmic integral function; the expression {{math|li(x^ρ)}} in the second term should be considered as {{math|Ei(ρ log x)}}, where {{math|Ei}} is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals. The final integral is equal to the series over the trivial zeros:

:$\backslash int\_x^\backslash infty\; \backslash frac\{\backslash mathrm\; dt\}\{t\; \backslash left(t^2\; -\; 1\backslash right)\; \backslash log\; t\}=\backslash int\_x^\backslash infty\; \backslash frac\{1\}\{t\backslash log\; t\}$

\left(\sum_{m}t^{-2m}\right)\,\mathrm dt=\sum_{m}\int_x^\infty \frac{t^{-2m}}{t\log t}

\,\mathrm dt \,\,\overset{\left(u=t^{-2m}\right)}{=}-\sum_{m} \operatorname{li}\left(x^{-2m}\right)

Thus, Möbius inversion formula gives us

:$\backslash pi\_0(x)\; =\; \backslash operatorname\{R\}(x)\; -\; \backslash sum\_\{\backslash rho\}\backslash operatorname\{R\}\backslash left(x^\backslash rho\backslash right)\; -\; \backslash sum\_\{m\}\; \backslash operatorname\{R\}\backslash left(x^\{-2m\}\backslash right)$

valid for {{math|x > 1}}, where

:$\backslash operatorname\{R\}(x)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{\backslash mu(n)\}\{n\}\; \backslash operatorname\{li\}\backslash left(x^\{1/n\}\backslash right)\; =\; 1\; +\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{\backslash left(\backslash log\; x\backslash right)^k\}\{k!\; k\; \backslash zeta(k+1)\}$

is Riemann's R-function{{MathWorld |title=Riemann Prime Counting Function |urlname=RiemannPrimeCountingFunction}} and {{math|μ(n)}} is the Möbius function. The latter series for it is known as Gram series.{{cite book | title=Prime Numbers and Computer Methods for Factorization | edition=2nd | first=Hans | last=Riesel | author-link=Hans Riesel | series=Progress in Mathematics | volume=126 | publisher=Birkhäuser | year=1994 | isbn=0-8176-3743-5 | pages=50–51 }}{{MathWorld |title=Gram Series |urlname=GramSeries}} Because {{math|log x < x}} for all {{math|x > 0}}, this series converges for all positive {{mvar|x}} by comparison with the series for {{mvar|e^x}}. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as {{math|ρ log x}} and not {{math|log x^ρ}}.

Folkmar Bornemann proved,{{cite web |last=Bornemann | first=Folkmar |title=Solution of a Problem Posed by Jörg Waldvogel |url=http://www-m3.ma.tum.de/bornemann/RiemannRZero.pdf }} when assuming the conjecture that all zeros of the Riemann zeta function are simple,Montgomery showed that (assuming the Riemann hypothesis) at least two thirds of all zeros are simple. that

:$\backslash operatorname\{R\}\backslash left(e^\{-2\backslash pi\; t\}\backslash right)=\backslash frac\{1\}\{\backslash pi\}\backslash sum\_\{k=1\}^\backslash infty\backslash frac\{(-1)^\{k-1\}t^\{-2k-1\}\}\{(2k+1)\backslash zeta(2k+1)\}+\backslash frac12\backslash sum\_\{\backslash rho\}\backslash frac\{t^\{-\backslash rho\}\}\{\backslash rho\backslash cos\backslash frac\{\backslash pi\backslash rho\}\{2\}\backslash zeta\text{'}(\backslash rho)\}$

where {{mvar|ρ}} runs over the non-trivial zeros of the Riemann zeta function and {{math|t > 0}}.

The sum over non-trivial zeta zeros in the formula for {{math|π₀(x)}} describes the fluctuations of {{math|π₀(x)}} while the remaining terms give the "smooth" part of prime-counting function,{{cite web |title=The encoding of the prime distribution by the zeta zeros |url=http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding1.htm |publisher=Matthew Watkins |access-date=2008-09-14}} so one can use

:$\backslash operatorname\{R\}(x)\; -\; \backslash sum\_\{m=1\}^\backslash infty\; \backslash operatorname\{R\}\backslash left(x^\{-2m\}\backslash right)$

as a good estimator of {{math|π(x)}} for {{math|x > 1}}. In fact, since the second term approaches 0 as {{math|x → ∞}}, while the amplitude of the "noisy" part is heuristically about {{math|{{sfrac|{{sqrt|x}}|log x}}}}, estimating {{math|π(x)}} by {{math|R(x)}} alone is just as good, and fluctuations of the distribution of primes may be clearly represented with the function

:$\backslash bigl(\; \backslash pi\_0(x)\; -\; \backslash operatorname\{R\}(x)\backslash bigr)\; \backslash frac\{\backslash log\; x\}\{\backslash sqrt\; x\}.$

Ramanujan{{Cite book|url=https://books.google.com/books?id=QoMHCAAAQBAJ|title=Ramanujan's Notebooks, Part IV|last=Berndt|first=Bruce C.|date=2012-12-06|pages=112–113|publisher=Springer Science & Business Media|isbn=9781461269328|language=en}} proved that the inequality

:

holds for all sufficiently large values of {{mvar|x}}.

Here are some useful inequalities for {{math|π(x)}}.

:

The left inequality holds for {{math|x ≥ 17}} and the right inequality holds for {{math|x > 1}}. The constant {{#expr:30*ln(113)/113 round 5}} is {{math|30{{sfrac|log 113|113}}}} to 5 decimal places, as {{math|π(x) {{sfrac|log x|x}}}} has its maximum value at {{math|1=x = p₃₀ = 113}}.{{Cite journal | author-link = J. Barkley Rosser | last1 = Rosser | first1 = J. Barkley | last2 = Schoenfeld | first2 = Lowell | title = Approximate formulas for some functions of prime numbers | journal = Illinois J. Math. | year = 1962 | volume = 6 | pages = 64–94 | doi = 10.1215/ijm/1255631807 | zbl = 0122.05001 | issn = 0019-2082 | url = https://projecteuclid.org/euclid.ijm/1255631807 | doi-access = free }}

Pierre Dusart proved in 2010:{{cite arXiv |last = Dusart |first = Pierre |author-link = Pierre Dusart |eprint=1002.0442v1 |title = Estimates of Some Functions Over Primes without R.H. |class = math.NT |date = 2 Feb 2010 }}

:

More recently, Dusart has proved{{cite journal |last = Dusart |first = Pierre |author-link = Pierre Dusart |title = Explicit estimates of some functions over primes |journal = Ramanujan Journal |volume = 45 |issue = 1 |pages=225–234 |date = January 2018 |doi = 10.1007/s11139-016-9839-4|s2cid = 125120533 }}

(Theorem 5.1) that

:$\backslash frac\{x\}\{\backslash log\; x\}\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{\backslash log\; x\}\; +\; \backslash frac\{2\}\{\backslash log^2\; x\}\; \backslash right)\; \backslash le\; \backslash pi(x)\; \backslash le\; \backslash frac\{x\}\{\backslash log\; x\}\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{\backslash log\; x\}\; +\; \backslash frac\{2\}\{\backslash log^2\; x\}\; +\; \backslash frac\{7.59\}\{\backslash log^3\; x\}\; \backslash right),$

for {{math|x ≥ 88789}} and {{math|x > 1}}, respectively.

Going in the other direction, an approximation for the {{mvar|n}}th prime, {{mvar|p_n}}, is

:$p\_n\; =\; n\; \backslash left(\backslash log\; n\; +\; \backslash log\backslash log\; n\; -\; 1\; +\; \backslash frac\; \{\backslash log\backslash log\; n\; -\; 2\}\{\backslash log\; n\}\; +\; O\backslash left(\; \backslash frac\; \{(\backslash log\backslash log\; n)^2\}\; \{(\backslash log\; n)^2\}\backslash right)\backslash right).$

Here are some inequalities for the {{mvar|n}}th prime. The lower bound is due to Dusart (1999){{cite journal

| author-link=Pierre Dusart

| last = Dusart

| first = Pierre

| date = January 1999

| title = The k^th prime is greater than k(ln k + ln ln k − 1) for k ≥ 2

| journal = Mathematics of Computation

| volume = 68

| issue = 225

| pages = 411–415

| doi = 10.1090/S0025-5718-99-01037-6

| doi-access = free

| bibcode = 1999MaCom..68..411D

| url = https://www.ams.org/mcom/1999-68-225/S0025-5718-99-01037-6/S0025-5718-99-01037-6.pdf

}} and the upper bound to Rosser (1941).{{cite journal

| first = Barkley | last = Rosser | author-link = J. Barkley Rosser

| date = January 1941

| title = Explicit bounds for some functions of prime numbers

| jstor = 2371291

| journal = American Journal of Mathematics

| volume = 63

| issue = 1

| pages = 211–232

| doi = 10.2307/2371291

}}

:

The left inequality holds for {{math|n ≥ 2}} and the right inequality holds for {{math|n ≥ 6}}. A variant form sometimes seen substitutes $\backslash log\; n\; +\backslash log\backslash log\; n\; =\; \backslash log(n\; \backslash log\; n).$ An even simpler lower bound is{{cite journal

| title = Approximate formulas for some functions of prime numbers

| first1 = J. Barkley | last1 = Rosser | author1-link = J. Barkley Rosser

| first2 = Lowell | last2 = Schoenfeld | author2-link = Lowell Schoenfeld

| journal = Illinois Journal of Mathematics

| volume = 6

| issue = 1

| pages = 64–94

| date = March 1962

| doi = 10.1215/ijm/1255631807

}}

:

which holds for all {{math|n ≥ 1}}, but the lower bound above is tighter for {{math|n > e^e ≈{{#expr:exp(exp(1)) round 3}}}}.

In 2010 Dusart proved (Propositions 6.7 and 6.6) that

:$n\; \backslash left(\; \backslash log\; n\; +\; \backslash log\; \backslash log\; n\; -\; 1\; +\; \backslash frac\{\backslash log\; \backslash log\; n\; -\; 2.1\}\{\backslash log\; n\}\; \backslash right)\; \backslash le\; p\_n\; \backslash le\; n\; \backslash left(\; \backslash log\; n\; +\; \backslash log\; \backslash log\; n\; -\; 1\; +\; \backslash frac\{\backslash log\; \backslash log\; n\; -\; 2\}\{\backslash log\; n\}\; \backslash right),$

for {{math|n ≥ 3}} and {{math|n ≥ 688383}}, respectively.

In 2024, Axler{{cite journal

| title = New estimates for the nth prime number

| first = Christian | last = Axler

| journal = Journal of Integer Sequences

| volume = 19

| issue = 4

| article-number = 2

| date = 2019

| orig-date = 23 Mar 2017

| arxiv = 1706.03651

| url = https://cs.uwaterloo.ca/journals/JIS/VOL22/Axler/axler17.html

}} further tightened this (equations 1.12 and 1.13) using bounds of the form

:$f(n,g(w))\; =\; n\; \backslash left(\; \backslash log\; n\; +\; \backslash log\backslash log\; n\; -\; 1\; +\; \backslash frac\{\backslash log\backslash log\; n\; -\; 2\}\{\backslash log\; n\}\; -\; \backslash frac\{g(\backslash log\backslash log\; n)\}\{2\backslash log^2\; n\}\; \backslash right)$

proving that

:$f(n,\; w^2\; -\; 6w\; +\; 11.321)\; \backslash le\; p\_n\; \backslash le\; f(n,\; w^2\; -\; 6w)$

for {{math|n ≥ 2}} and {{math|n ≥ 3468}}, respectively.

The lower bound may also be simplified to {{math|f(n, w²)}} without altering its validity. The upper bound may be tightened to {{math|f(n, w² − 6w + 10.667)}} if {{math|n ≥ 46254381}}.

There are additional bounds of varying complexity.{{cite web

| title = Bounds for n-th prime

| url = https://math.stackexchange.com/questions/1270814/bounds-for-n-th-prime

| date = 31 December 2015

| website = Mathematics StackExchange

}}{{cite journal

| title = New Estimates for Some Functions Defined Over Primes

| first = Christian | last = Axler

| journal = Integers

| volume = 18

| article-number = A52

| doi = 10.5281/zenodo.10677755

| doi-access = free

| date = 2018

| orig-date = 23 Mar 2017

| arxiv = 1703.08032

| url = https://math.colgate.edu/~integers/s52/s52.pdf

}}{{cite journal

| title = Effective Estimates for Some Functions Defined over Primes

| first = Christian | last = Axler

| journal = Integers

| volume = 24

| article-number = A34

| doi = 10.5281/zenodo.10677755

| doi-access = free

| date = 2024

| orig-date = 11 Mar 2022

| arxiv = 2203.05917

| url = https://math.colgate.edu/~integers/y34/y34.pdf

}}

The Riemann hypothesis implies a much tighter bound on the error in the estimate for {{math|π(x)}}, and hence to a more regular distribution of prime numbers,

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

Specifically,{{Cite journal | last1=Schoenfeld | first1=Lowell |author-link=Lowell Schoenfeld| title=Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II | doi=10.2307/2005976 | mr=0457374 | year=1976 | journal=Mathematics of Computation | issn=0025-5718 | volume=30 | issue=134 | pages=337–360 | jstor=2005976 | publisher=American Mathematical Society}}

:

{{harvtxt|Dudek|2015}} proved that the Riemann hypothesis implies that for all {{math|x ≥ 2}} there is a prime {{mvar|p}} satisfying

:

• Bertrand's postulate
• Oppermann's conjecture
• Ramanujan prime

{{Reflist}}

{{reflist|group=note}}

• Chris Caldwell, [http://primes.utm.edu/nthprime/ The Nth Prime Page] at The Prime Pages.
• Tomás Oliveira e Silva, [http://sweet.ua.pt/tos/primes.html Tables of prime-counting functions].
• {{Citation| last=Dudek|first=Adrian W.|date=2015|title=On the Riemann hypothesis and the difference between primes|journal=International Journal of Number Theory|volume=11|issue=3|pages=771–778|doi=10.1142/S1793042115500426|issn=1793-0421|arxiv=1402.6417|bibcode=2014arXiv1402.6417D|s2cid=119321107}}

{{DEFAULTSORT:Prime-Counting Function}}

Category:Analytic number theory

Category:Prime numbers

Category:Arithmetic functions