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<title>Prime-counting function</title>

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Prime-counting function

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</style><div role="note" class="hatnote navigation-not-searchable">"Π(x)" redirects here. For the variant of the gamma function, see <a href="Gamma_function#Pi_function" title="Gamma function">Gamma function § Pi function</a>.</div>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the prime-counting function is the <a href="Function_(mathematics)" title="Function (mathematics)">function</a> counting the number of <a href="Prime_number" title="Prime number">prime numbers</a> less than or equal to some <a href="Real_number" title="Real number">real number</a> x.<a href="#cite_note-Bach-1">[1]</a><a href="#cite_note-mathworld_pcf-2">[2]</a> It is denoted by π(x) (unrelated to the <a href="Pi" title="Pi">number π</a>).

A symmetric variant seen sometimes is π0(x), which is equal to π(x) − <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style>1⁄2 if x is exactly a prime number, and equal to π(x) otherwise. That is, the number of prime numbers less than x, plus half if x equals a prime.

<div class="mw-heading mw-heading2"><h2 id="Growth_rate">Growth rate</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Prime_number_theorem" title="Prime number theorem">Prime number theorem</a></div>

Of great interest in <a href="Number_theory" title="Number theory">number theory</a> is the <a href="Asymptotic_analysis" title="Asymptotic analysis">growth rate</a> of the prime-counting function.<a href="#cite_note-Caldwell-3">[3]</a><a href="#cite_note-Dickson-4">[4]</a> It was <a href="Conjecture" title="Conjecture">conjectured</a> in the end of the 18th century by <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> and by <a href="Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> to be approximately

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{\log x}}}</annotation>

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where log is the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>, in the sense that

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\log x}}=1.}</annotation>

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This statement is the <a href="Prime_number_theorem" title="Prime number theorem">prime number theorem</a>. An equivalent statement is

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{\operatorname {li} (x)}}=1}</annotation>

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</math>

where li is the <a href="Logarithmic_integral" class="mw-redirect" title="Logarithmic integral">logarithmic integral</a> function. The prime number theorem was first proved in 1896 by <a href="Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a> and by <a href="Charles_Jean_de_la_Vall%C3%A9e-Poussin" class="mw-redirect" title="Charles Jean de la Vallée-Poussin">Charles de la Vallée Poussin</a> independently, using properties of the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> introduced by <a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a> in 1859. Proofs of the prime number theorem not using the zeta function or <a href="Complex_analysis" title="Complex analysis">complex analysis</a> were found around 1948 by <a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a> and by <a href="Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> (for the most part independently).<a href="#cite_note-Ireland-5">[5]</a>

<div class="mw-heading mw-heading3"><h3 id="More_precise_estimates">More precise estimates</h3></div>

In 1899, <a href="Charles_Jean_de_la_Vall%C3%A9e_Poussin" title="Charles Jean de la Vallée Poussin">de la Vallée Poussin</a> proved that

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<annotation encoding="application/x-tex">{\displaystyle \pi (x)=\operatorname {li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }</annotation>

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for some positive constant a. Here, O(...) is the <a href="Big_O_notation" title="Big O notation">big O notation</a>.

More precise estimates of π(x) are now known. For example, in 2002, <a href="Kevin_Ford_(mathematician)" title="Kevin Ford (mathematician)">Kevin Ford</a> proved that<a href="#cite_note-Ford-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle \pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-0.2098(\log x)^{3/5}(\log \log x)^{-1/5}\right)\right).}</annotation>

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Mossinghoff and <a href="Timothy_Trudgian" title="Timothy Trudgian">Trudgian</a> proved<a href="#cite_note-8">[8]</a> an explicit upper bound for the difference between π(x) and li(x):

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<annotation encoding="application/x-tex">{\displaystyle {\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.}</annotation>

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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 <a href="Skewes'_number" class="mw-redirect" title="Skewes' number">Skewes' number</a>.

<div class="mw-heading mw-heading3"><h3 id="Exact_form">Exact form</h3></div>

For x > 1 let π0(x) = π(x) − <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/2⁠ when x is a prime number, and π0(x) = π(x) otherwise. <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>, in his work <a href="On_the_Number_of_Primes_Less_Than_a_Given_Magnitude" title="On the Number of Primes Less Than a Given Magnitude">On the Number of Primes Less Than a Given Magnitude</a>, proved that π0(x) is equal to<a href="#cite_note-9">[9]</a>

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<annotation encoding="application/x-tex">{\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho }),}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} \left(x^{1/n}\right),}</annotation>

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</math>

μ(n) is the <a href="M%C3%B6bius_function" title="Möbius function">Möbius function</a>, li(x) is the <a href="Logarithmic_integral_function" title="Logarithmic integral function">logarithmic integral function</a>, ρ indexes every zero of the Riemann zeta function, and li(x⁠ρ/n⁠) is not evaluated with a <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a> but instead considered as Ei(⁠ρ/n⁠ log x) where Ei(x) is the <a href="Exponential_integral" title="Exponential integral">exponential integral</a>. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then π0(x) may be approximated by<a href="#cite_note-RieselGohl-10">[10]</a>

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<mi>arctan</mi>

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<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

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</math>

The <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> suggests that every such non-trivial zero lies along Re(s) = ⁠1/2⁠.

<div class="mw-heading mw-heading2"><h2 id="Table_of_π(x),_⁠x/log_x_⁠,_and_li(x)">Table of π(x), ⁠x/log x ⁠, and li(x)</h2></div>

The table shows how the three functions π(x), ⁠x/log x⁠, and li(x) compared at powers of 10. See also,<a href="#cite_note-Caldwell-3">[3]</a><a href="#cite_note-Silva-11">[11]</a> and<a href="#cite_note-Gourdon-12">[12]</a>

<th>x

</th>

<th>π(x)

</th>

<th>π(x) − ⁠x/log x⁠

</th>

<th>li(x) − π(x)

</th>

<th>⁠x/π(x)⁠

</th>

<th>⁠x/log x⁠  % error

</th></tr>

<tr>

<td>10

</td>

<td>4

</td>

<td>0

</td>

<td>2

</td>

<td>2.500

</td>

<td>−8.57%

</td></tr>

<tr>

<td>102

</td>

<td>25

</td>

<td>3

</td>

<td>5

</td>

<td>4.000

</td>

<td>+13.14%

</td></tr>

<tr>

<td>103

</td>

<td>168

</td>

<td>23

</td>

<td>10

</td>

<td>5.952

</td>

<td>+13.83%

</td></tr>

<tr>

<td>104

</td>

<td>1,229

</td>

<td>143

</td>

<td>17

</td>

<td>8.137

</td>

<td>+11.66%

</td></tr>

<tr>

<td>105

</td>

<td>9,592

</td>

<td>906

</td>

<td>38

</td>

<td>10.425

</td>

<td>+9.45%

</td></tr>

<tr>

<td>106

</td>

<td>78,498

</td>

<td>6,116

</td>

<td>130

</td>

<td>12.739

</td>

<td>+7.79%

</td></tr>

<tr>

<td>107

</td>

<td>664,579

</td>

<td>44,158

</td>

<td>339

</td>

<td>15.047

</td>

<td>+6.64%

</td></tr>

<tr>

<td>108

</td>

<td>5,761,455

</td>

<td>332,774

</td>

<td>754

</td>

<td>17.357

</td>

<td>+5.78%

</td></tr>

<tr>

<td>109

</td>

<td>50,847,534

</td>

<td>2,592,592

</td>

<td>1,701

</td>

<td>19.667

</td>

<td>+5.10%

</td></tr>

<tr>

<td>1010

</td>

<td>455,052,511

</td>

<td>20,758,029

</td>

<td>3,104

</td>

<td>21.975

</td>

<td>+4.56%

</td></tr>

<tr>

<td>1011

</td>

<td>4,118,054,813

</td>

<td>169,923,159

</td>

<td>11,588

</td>

<td>24.283

</td>

<td>+4.13%

</td></tr>

<tr>

<td>1012

</td>

<td>37,607,912,018

</td>

<td>1,416,705,193

</td>

<td>38,263

</td>

<td>26.590

</td>

<td>+3.77%

</td></tr>

<tr>

<td>1013

</td>

<td>346,065,536,839

</td>

<td>11,992,858,452

</td>

<td>108,971

</td>

<td>28.896

</td>

<td>+3.47%

</td></tr>

<tr>

<td>1014

</td>

<td>3,204,941,750,802

</td>

<td>102,838,308,636

</td>

<td>314,890

</td>

<td>31.202

</td>

<td>+3.21%

</td></tr>

<tr>

<td>1015

</td>

<td>29,844,570,422,669

</td>

<td>891,604,962,452

</td>

<td>1,052,619

</td>

<td>33.507

</td>

<td>+2.99%

</td></tr>

<tr>

<td>1016

</td>

<td>279,238,341,033,925

</td>

<td>7,804,289,844,393

</td>

<td>3,214,632

</td>

<td>35.812

</td>

<td>+2.79%

</td></tr>

<tr>

<td>1017

</td>

<td>2,623,557,157,654,233

</td>

<td>68,883,734,693,928

</td>

<td>7,956,589

</td>

<td>38.116

</td>

<td>+2.63%

</td></tr>

<tr>

<td>1018

</td>

<td>24,739,954,287,740,860

</td>

<td>612,483,070,893,536

</td>

<td>21,949,555

</td>

<td>40.420

</td>

<td>+2.48%

</td></tr>

<tr>

<td>1019

</td>

<td>234,057,667,276,344,607

</td>

<td>5,481,624,169,369,961

</td>

<td>99,877,775

</td>

<td>42.725

</td>

<td>+2.34%

</td></tr>

<tr>

<td>1020

</td>

<td>2,220,819,602,560,918,840

</td>

<td>49,347,193,044,659,702

</td>

<td>222,744,644

</td>

<td>45.028

</td>

<td>+2.22%

</td></tr>

<tr>

<td>1021

</td>

<td>21,127,269,486,018,731,928

</td>

<td>446,579,871,578,168,707

</td>

<td>597,394,254

</td>

<td>47.332

</td>

<td>+2.11%

</td></tr>

<tr>

<td>1022

</td>

<td>201,467,286,689,315,906,290

</td>

<td>4,060,704,006,019,620,994

</td>

<td>1,932,355,208

</td>

<td>49.636

</td>

<td>+2.02%

</td></tr>

<tr>

<td>1023

</td>

<td>1,925,320,391,606,803,968,923

</td>

<td>37,083,513,766,578,631,309

</td>

<td>7,250,186,216

</td>

<td>51.939

</td>

<td>+1.93%

</td></tr>

<tr>

<td>1024

</td>

<td>18,435,599,767,349,200,867,866

</td>

<td>339,996,354,713,708,049,069

</td>

<td>17,146,907,278

</td>

<td>54.243

</td>

<td>+1.84%

</td></tr>

<tr>

<td>1025

</td>

<td>176,846,309,399,143,769,411,680

</td>

<td>3,128,516,637,843,038,351,228

</td>

<td>55,160,980,939

</td>

<td>56.546

</td>

<td>+1.77%

</td></tr>

<tr>

<td>1026

</td>

<td>1,699,246,750,872,437,141,327,603

</td>

<td>28,883,358,936,853,188,823,261

</td>

<td>155,891,678,121

</td>

<td>58.850

</td>

<td>+1.70%

</td></tr>

<tr>

<td>1027

</td>

<td>16,352,460,426,841,680,446,427,399

</td>

<td>267,479,615,610,131,274,163,365

</td>

<td>508,666,658,006

</td>

<td>61.153

</td>

<td>+1.64%

</td></tr>

<tr>

<td>1028

</td>

<td>157,589,269,275,973,410,412,739,598

</td>

<td>2,484,097,167,669,186,251,622,127

</td>

<td>1,427,745,660,374

</td>

<td>63.456

</td>

<td>+1.58%

</td></tr>

<tr>

<td>1029

</td>

<td>1,520,698,109,714,272,166,094,258,063

</td>

<td>23,130,930,737,541,725,917,951,446

</td>

<td>4,551,193,622,464

</td>

<td>65.759

</td>

<td>+1.52%

</td></tr></tbody></table></dd></dl>

In the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>, the π(x) column is sequence <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A006880" class="extiw external" title="oeis:A006880">A006880</a>, π(x) − ⁠x/log x⁠ is sequence <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A057835" class="extiw external" title="oeis:A057835">A057835</a>, and li(x) − π(x) is sequence <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A057752" class="extiw external" title="oeis:A057752">A057752</a>.

The value for π(1024) was originally computed by J. Buethe, <a href="Jens_Franke" title="Jens Franke">J. Franke</a>, A. Jost, and T. Kleinjung assuming the <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>.<a href="#cite_note-Franke-13">[13]</a>

It was later verified unconditionally in a computation by D. J. Platt.<a href="#cite_note-Platt2012-14">[14]</a>

The value for π(1025) is by the same four authors.<a href="#cite_note-Buethe-15">[15]</a>

The value for π(1026) was computed by D. B. Staple.<a href="#cite_note-Staple-16">[16]</a> 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,<a href="#cite_note-17">[17]</a> 2020,<a href="#cite_note-18">[18]</a> and 2022,<a href="#cite_note-19">[19]</a> respectively.

<div class="mw-heading mw-heading2"><h2 id="Algorithms_for_evaluating_π(x)">Algorithms for evaluating π(x)</h2></div>

A simple way to find π(x), if x is not too large, is to use the <a href="Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">sieve of Eratosthenes</a> to produce the primes less than or equal to x and then to count them.

A more elaborate way of finding π(x) is due to <a href="Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> (using the <a href="Inclusion%E2%80%93exclusion_principle" title="Inclusion–exclusion principle">inclusion–exclusion principle</a>): given x, if p1, p2,…, pn are distinct prime numbers, then the number of integers less than or equal to x which are divisible by no pi is

</mrow>

</munder>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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 }</annotation>

</semantics>

</math><img src="./28855ea8083ed7bc59b216a71227c3d9c8b26cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.972ex; height:7.009ex;" alt="{\displaystyle \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 }" loading="lazy"></dd></dl>

(where ⌊x⌋ denotes the <a href="Floor_function" class="mw-redirect" title="Floor function">floor function</a>). This number is therefore equal to

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (x)-\pi \left({\sqrt {x}}\right)+1}</annotation>

</semantics>

</math><img src="./35658691efbc18be1b334587dd526341b1f7171b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.108ex; height:3.009ex;" alt="{\displaystyle \pi (x)-\pi \left({\sqrt {x}}\right)+1}" loading="lazy"></dd></dl>

when the numbers p1, p2,…, pn are the prime numbers less than or equal to the <a href="Square_root" title="Square root">square root</a> of x.

<div class="mw-heading mw-heading3"><h3 id="The_Meissel–Lehmer_algorithm">The Meissel–Lehmer algorithm</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Meissel%E2%80%93Lehmer_algorithm" title="Meissel–Lehmer algorithm">Meissel–Lehmer algorithm</a></div>

In a series of articles published between 1870 and 1885, <a href="Ernst_Meissel" title="Ernst Meissel">Ernst Meissel</a> described (and used) a practical combinatorial way of evaluating π(x): Let p1, p2,…, pn be the first n primes and denote by Φ(m,n) the number of natural numbers not greater than m which are divisible by none of the pi for any i ≤ n. Then

<mrow>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (m,n)=\Phi (m,n-1)-\Phi \left({\frac {m}{p_{n}}},n-1\right).}</annotation>

</semantics>

</math><img src="./463121ee021bdd520b7cc2f948ced7429df5af01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.03ex; height:6.176ex;" alt="{\displaystyle \Phi (m,n)=\Phi (m,n-1)-\Phi \left({\frac {m}{p_{n}}},n-1\right).}" loading="lazy"></dd></dl>

Given a natural number m, if n = π(3√m) and if μ = π(√m) − n, then

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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).}</annotation>

</semantics>

</math><img src="./49dae2e94d6979ea0cd9c9978e56fbcbe54cb311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.179ex; height:7.009ex;" alt="{\displaystyle \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).}" loading="lazy"></dd></dl>

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

In 1959, <a href="Derrick_Henry_Lehmer" class="mw-redirect" title="Derrick Henry Lehmer">Derrick Henry Lehmer</a> extended and simplified Meissel's method. Define, for real m and for natural numbers n and k, Pk(m,n) as the number of numbers not greater than m with exactly k prime factors, all greater than pn. Furthermore, set P0(m,n) = 1. Then

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (m,n)=\sum _{k=0}^{+\infty }P_{k}(m,n)}</annotation>

</semantics>

</math><img src="./118229d62542d234f8714a4b90bc159234de1ead.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.656ex; height:7.343ex;" alt="{\displaystyle \Phi (m,n)=\sum _{k=0}^{+\infty }P_{k}(m,n)}" loading="lazy"></dd></dl>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (m)=\Phi (m,n)+n-1-P_{2}(m,n)}</annotation>

</semantics>

</math><img src="./1fe53cdf8ad6f08bdcf2095e3cace38e82be89d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.14ex; height:2.843ex;" alt="{\displaystyle \pi (m)=\Phi (m,n)+n-1-P_{2}(m,n)}" loading="lazy"></dd></dl>

The computation of P2(m,n) can be obtained this way:

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</munder>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{2}(m,n)=\sum _{y<p\leq {\sqrt {m}}}\left(\pi \left({\frac {m}{p}}\right)-\pi (p)+1\right)}</annotation>

</semantics>

</math><img src="./104b24bf3c4c57f139c5c527051370e5e610cd75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:41.915ex; height:7.343ex;" alt="{\displaystyle P_{2}(m,n)=\sum _{y<p\leq {\sqrt {m}}}\left(\pi \left({\frac {m}{p}}\right)-\pi (p)+1\right)}" loading="lazy"></dd></dl>

where the sum is over prime numbers.

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

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (m,0)=\lfloor m\rfloor }</annotation>

</semantics>

</math><img src="./b55ef167806979b8a3a239262203e582ff4e4987.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.928ex; height:2.843ex;" alt="{\displaystyle \Phi (m,0)=\lfloor m\rfloor }" loading="lazy"></li>

<mrow>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (m,b)=\Phi (m,b-1)-\Phi \left({\frac {m}{p_{b}}},b-1\right)}</annotation>

</semantics>

</math><img src="./4de3cc5e77ebf21791f0f860120e4590635876a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.524ex; height:6.176ex;" alt="{\displaystyle \Phi (m,b)=\Phi (m,b-1)-\Phi \left({\frac {m}{p_{b}}},b-1\right)}" loading="lazy"></li></ol>

Using his method and an <a href="IBM_701" title="IBM 701">IBM 701</a>, Lehmer was able to compute the correct value of π(109) and missed the correct value of π(1010) by 1.<a href="#cite_note-lehmer-20">[20]</a>

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.<a href="#cite_note-pix_comp-21">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="Other_prime-counting_functions">Other prime-counting functions</h2></div>

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

<div class="mw-heading mw-heading3"><h3 id="Riemann's_prime-power_counting_function">Riemann's prime-power counting function</h3></div>

Riemann's prime-power counting function is usually denoted as Π0(x) or J0(x). It has jumps of ⁠1/n⁠ at prime powers pn and it takes a value halfway between the two sides at the discontinuities of π(x). That added detail is used because the function may then be defined by an inverse <a href="Mellin_transform" title="Mellin transform">Mellin transform</a>.

Formally, we may define Π0(x) by

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi _{0}(x)={\frac {1}{2}}\left(\sum _{p^{n}<x}{\frac {1}{n}}+\sum _{p^{n}\leq x}{\frac {1}{n}}\right)\ }</annotation>

</semantics>

</math><img src="./160accb459972018a71929c40b5075f16601841a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:32.167ex; height:7.843ex;" alt="{\displaystyle \Pi _{0}(x)={\frac {1}{2}}\left(\sum _{p^{n}<x}{\frac {1}{n}}+\sum _{p^{n}\leq x}{\frac {1}{n}}\right)\ }" loading="lazy"></dd></dl>

where the variable p in each sum ranges over all primes within the specified limits.

We may also write

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ \Pi _{0}(x)=\sum _{n=2}^{x}{\frac {\Lambda (n)}{\log n}}-{\frac {\Lambda (x)}{2\log x}}=\sum _{n=1}^{\infty }{\frac {1}{n}}\pi _{0}\left(x^{1/n}\right)}</annotation>

</semantics>

</math><img src="./af0099cba3cb18057a0af4881865bde00f33bb32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.731ex; height:6.843ex;" alt="{\displaystyle \ \Pi _{0}(x)=\sum _{n=2}^{x}{\frac {\Lambda (n)}{\log n}}-{\frac {\Lambda (x)}{2\log x}}=\sum _{n=1}^{\infty }{\frac {1}{n}}\pi _{0}\left(x^{1/n}\right)}" loading="lazy"></dd></dl>

where Λ is the <a href="Von_Mangoldt_function" title="Von Mangoldt function">von Mangoldt function</a> and

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\pi (x-\varepsilon )+\pi (x+\varepsilon )}{2}}.}</annotation>

</semantics>

</math><img src="./20a40a5bdb8805b86a96b61511525769d36d8da6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.348ex; height:5.843ex;" alt="{\displaystyle \pi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\pi (x-\varepsilon )+\pi (x+\varepsilon )}{2}}.}" loading="lazy"></dd></dl>

The <a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion formula</a> then gives

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\ \Pi _{0}\left(x^{1/n}\right),}</annotation>

</semantics>

</math><img src="./9904e72c2c4dd34408f732853d88d10384bf9086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.567ex; height:6.843ex;" alt="{\displaystyle \pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\ \Pi _{0}\left(x^{1/n}\right),}" loading="lazy"></dd></dl>

where μ(n) is the <a href="M%C3%B6bius_function" title="Möbius function">Möbius function</a>.

Knowing the relationship between the logarithm of the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> and the <a href="Von_Mangoldt_function" title="Von Mangoldt function">von Mangoldt function</a> Λ, and using the <a href="Perron_formula" class="mw-redirect" title="Perron formula">Perron formula</a> we have

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)x^{-s-1}\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./2af5fdbfa073b9bf4f880ab1f13611d30d9451cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.314ex; height:5.843ex;" alt="{\displaystyle \log \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)x^{-s-1}\,\mathrm {d} x}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Chebyshev's_function">Chebyshev's function</h3></div>

The <a href="Chebyshev_function" title="Chebyshev function">Chebyshev function</a> weights primes or prime powers pn by log p:

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</munder>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\vartheta (x)&=\sum _{p\leq x}\log p\\\psi (x)&=\sum _{p^{n}\leq x}\log p=\sum _{n=1}^{\infty }\vartheta \left(x^{1/n}\right)=\sum _{n\leq x}\Lambda (n).\end{aligned}}}</annotation>

</semantics>

</math><img src="./41a2b156eac5f551c06e37e080c215a86d008150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.793ex; margin-bottom: -0.212ex; width:45.302ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\vartheta (x)&=\sum _{p\leq x}\log p\\\psi (x)&=\sum _{p^{n}\leq x}\log p=\sum _{n=1}^{\infty }\vartheta \left(x^{1/n}\right)=\sum _{n\leq x}\Lambda (n).\end{aligned}}}" loading="lazy"></dd></dl>

For x ≥ 2,<a href="#cite_note-22">[22]</a>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vartheta (x)=\pi (x)\log x-\int _{2}^{x}{\frac {\pi (t)}{t}}\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./9f9894770140df494589a221a2b4413e0593d606.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.358ex; height:6.176ex;" alt="{\displaystyle \vartheta (x)=\pi (x)\log x-\int _{2}^{x}{\frac {\pi (t)}{t}}\,\mathrm {d} t}" loading="lazy"></dd></dl>

and

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (x)={\frac {\vartheta (x)}{\log x}}+\int _{2}^{x}{\frac {\vartheta (t)}{t\log ^{2}(t)}}\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./48b9e8fa405f84183841dcddd20ba4db8689c821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.475ex; height:6.843ex;" alt="{\displaystyle \pi (x)={\frac {\vartheta (x)}{\log x}}+\int _{2}^{x}{\frac {\vartheta (t)}{t\log ^{2}(t)}}\mathrm {d} t.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Formulas_for_prime-counting_functions">Formulas for prime-counting functions</h2></div>

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 <a href="Prime_number_theorem" title="Prime number theorem">prime number theorem</a>. They stem from the work of Riemann and <a href="Hans_Carl_Friedrich_von_Mangoldt" title="Hans Carl Friedrich von Mangoldt">von Mangoldt</a>, and are generally known as <a href="Explicit_formulae_(L-function)" class="mw-redirect" title="Explicit formulae (L-function)">explicit formulae</a>.<a href="#cite_note-Titchmarsh-23">[23]</a>

We have the following expression for the second <a href="Chebyshev_function" title="Chebyshev function">Chebyshev function</a> ψ:

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log 2\pi -{\frac {1}{2}}\log \left(1-x^{-2}\right),}</annotation>

</semantics>

</math><img src="./d4791d3546b838dab3c728f4d74dd0f77c0ea65c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.298ex; height:6.676ex;" alt="{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log 2\pi -{\frac {1}{2}}\log \left(1-x^{-2}\right),}" loading="lazy"></dd></dl>

where

<msub>

</mrow>

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</munder>

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.}</annotation>

</semantics>

</math><img src="./c55122c67e4cd7e21200273b7ea72801ad7d2e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.899ex; height:5.843ex;" alt="{\displaystyle \psi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.}" loading="lazy"></dd></dl>

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 <a href="Absolute_value" title="Absolute value">absolute value</a> of the imaginary part. Note that the same sum over the trivial roots gives the last <a href="Subtrahend" class="mw-redirect" title="Subtrahend">subtrahend</a> in the formula.

For Π0(x) we have a more complicated formula

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<annotation encoding="application/x-tex">{\displaystyle \Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} \left(x^{\rho }\right)-\log 2+\int _{x}^{\infty }{\frac {\mathrm {d} t}{t\left(t^{2}-1\right)\log t}}.}</annotation>

</semantics>

</math><img src="./647589bbbfed457d1228c225eaa64e680446ea55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.495ex; height:6.843ex;" alt="{\displaystyle \Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} \left(x^{\rho }\right)-\log 2+\int _{x}^{\infty }{\frac {\mathrm {d} t}{t\left(t^{2}-1\right)\log t}}.}" loading="lazy"></dd></dl>

Again, the formula is valid for x > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value. The first term li(x) is the usual <a href="Logarithmic_integral_function" title="Logarithmic integral function">logarithmic integral function</a>; the expression li(xρ) in the second term should be considered as Ei(ρ log x), where Ei is the <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a> of the <a href="Exponential_integral" title="Exponential integral">exponential integral</a> function from negative reals to the <a href="Complex_plane" title="Complex plane">complex plane</a> with branch cut along the positive reals. The final integral is equal to the series over the trivial zeros:

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<annotation encoding="application/x-tex">{\displaystyle \int _{x}^{\infty }{\frac {\mathrm {d} t}{t\left(t^{2}-1\right)\log t}}=\int _{x}^{\infty }{\frac {1}{t\log t}}\left(\sum _{m}t^{-2m}\right)\,\mathrm {d} t=\sum _{m}\int _{x}^{\infty }{\frac {t^{-2m}}{t\log t}}\,\mathrm {d} t\,\,{\overset {\left(u=t^{-2m}\right)}{=}}-\sum _{m}\operatorname {li} \left(x^{-2m}\right)}</annotation>

</semantics>

</math><img src="./1bbf603a0f7fe6c473cd146998a1bb67b3b302ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:92.092ex; height:7.509ex;" alt="{\displaystyle \int _{x}^{\infty }{\frac {\mathrm {d} t}{t\left(t^{2}-1\right)\log t}}=\int _{x}^{\infty }{\frac {1}{t\log t}}\left(\sum _{m}t^{-2m}\right)\,\mathrm {d} t=\sum _{m}\int _{x}^{\infty }{\frac {t^{-2m}}{t\log t}}\,\mathrm {d} t\,\,{\overset {\left(u=t^{-2m}\right)}{=}}-\sum _{m}\operatorname {li} \left(x^{-2m}\right)}" loading="lazy"></dd></dl>

Thus, <a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion formula</a> gives us<a href="#cite_note-RieselGohl-10">[10]</a>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} \left(x^{\rho }\right)-\sum _{m}\operatorname {R} \left(x^{-2m}\right)}</annotation>

</semantics>

</math><img src="./e20f10eea2fe7818b9c516a8c38ed056ea710ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:41.509ex; height:5.843ex;" alt="{\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} \left(x^{\rho }\right)-\sum _{m}\operatorname {R} \left(x^{-2m}\right)}" loading="lazy"></dd></dl>

valid for x > 1, where

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} \left(x^{1/n}\right)=1+\sum _{k=1}^{\infty }{\frac {\left(\log x\right)^{k}}{k!k\zeta (k+1)}}}</annotation>

</semantics>

</math><img src="./b62ed345776a1ac5bb0a794071c8d80a71db2cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.648ex; height:7.343ex;" alt="{\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} \left(x^{1/n}\right)=1+\sum _{k=1}^{\infty }{\frac {\left(\log x\right)^{k}}{k!k\zeta (k+1)}}}" loading="lazy"></dd></dl>

is Riemann's R-function<a href="#cite_note-mathworld_r-24">[24]</a> and μ(n) is the <a href="M%C3%B6bius_function" title="Möbius function">Möbius function</a>. The latter series for it is known as <a href="J%C3%B8rgen_Pedersen_Gram" title="Jørgen Pedersen Gram">Gram</a> series.<a href="#cite_note-Riesel94-25">[25]</a><a href="#cite_note-mathworld_gram-26">[26]</a> Because log x < x for all x > 0, this series converges for all positive x by comparison with the series for ex. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as ρ log x and not log xρ.

Folkmar Bornemann proved,<a href="#cite_note-27">[27]</a> when assuming the <a href="Conjecture" title="Conjecture">conjecture</a> that all zeros of the Riemann zeta function are simple,<a href="#cite_note-28">[note 1]</a> that

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} \left(e^{-2\pi t}\right)={\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta (2k+1)}}+{\frac {1}{2}}\sum _{\rho }{\frac {t^{-\rho }}{\rho \cos {\frac {\pi \rho }{2}}\zeta '(\rho )}}}</annotation>

</semantics>

</math><img src="./ead1813878acfd98f90631d5db40c400c61fdb7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:58.825ex; height:7.343ex;" alt="{\displaystyle \operatorname {R} \left(e^{-2\pi t}\right)={\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta (2k+1)}}+{\frac {1}{2}}\sum _{\rho }{\frac {t^{-\rho }}{\rho \cos {\frac {\pi \rho }{2}}\zeta '(\rho )}}}" loading="lazy"></dd></dl>

where ρ runs over the non-trivial zeros of the Riemann zeta function and t > 0.

The sum over non-trivial zeta zeros in the formula for π0(x) describes the fluctuations of π0(x) while the remaining terms give the "smooth" part of prime-counting function,<a href="#cite_note-Watkins-29">[28]</a> so one can use

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {R} (x)-\sum _{m=1}^{\infty }\operatorname {R} \left(x^{-2m}\right)}</annotation>

</semantics>

</math><img src="./70c29051422ed59df201cee342e88b3e5848761b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.566ex; height:6.843ex;" alt="{\displaystyle \operatorname {R} (x)-\sum _{m=1}^{\infty }\operatorname {R} \left(x^{-2m}\right)}" loading="lazy"></dd></dl>

as a good estimator of π(x) for x > 1. In fact, since the second term approaches 0 as x → ∞, while the amplitude of the "noisy" part is heuristically about ⁠√x/log x⁠, estimating π(x) by R(x) alone is just as good, and fluctuations of the <a href="Distribution_of_primes" class="mw-redirect" title="Distribution of primes">distribution of primes</a> may be clearly represented with the function

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}\pi _{0}(x)-\operatorname {R} (x){\bigr )}{\frac {\log x}{\sqrt {x}}}.}</annotation>

</semantics>

</math><img src="./bdd8b2c09a89757b2fa74b4f9c5495d1c4227e31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.51ex; height:6.343ex;" alt="{\displaystyle {\bigl (}\pi _{0}(x)-\operatorname {R} (x){\bigr )}{\frac {\log x}{\sqrt {x}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Inequalities">Inequalities</h2></div>

<a href="Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Ramanujan</a><a href="#cite_note-30">[29]</a> proved that the inequality

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<annotation encoding="application/x-tex">{\displaystyle \pi (x)^{2}<{\frac {ex}{\log x}}\pi \left({\frac {x}{e}}\right)}</annotation>

</semantics>

</math><img src="./50a0d589fe787fc81945e91420ce8511617662b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.809ex; height:5.343ex;" alt="{\displaystyle \pi (x)^{2}<{\frac {ex}{\log x}}\pi \left({\frac {x}{e}}\right)}" loading="lazy"></dd></dl>

holds for all sufficiently large values of x.

Here are some useful inequalities for π(x).

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{\log x}}<\pi (x)<1.25506{\frac {x}{\log x}}\quad {\text{for }}x\geq 17.}</annotation>

</semantics>

</math><img src="./3f6e18fb91e5a9725995cdff553e9fe5dfe4c412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.428ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{\log x}}<\pi (x)<1.25506{\frac {x}{\log x}}\quad {\text{for }}x\geq 17.}" loading="lazy"></dd></dl>

The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is 30⁠log 113/113⁠ to 5 decimal places, as π(x) ⁠log x/x⁠ has its maximum value at x = p30 = 113.<a href="#cite_note-Rosser1962-31">[30]</a>

<a href="Pierre_Dusart" title="Pierre Dusart">Pierre Dusart</a> proved in 2010:<a href="#cite_note-Dusart2010-32">[31]</a>

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<mtext> respectively.</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{\log x-1}}<\pi (x)<{\frac {x}{\log x-1.1}}\quad {\text{for }}x\geq 5393{\text{ and }}x\geq 60184,{\text{ respectively.}}}</annotation>

</semantics>

</math><img src="./69492ba54c47a5e6b1be6ecaf24a01b7faf927e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:75.613ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{\log x-1}}<\pi (x)<{\frac {x}{\log x-1.1}}\quad {\text{for }}x\geq 5393{\text{ and }}x\geq 60184,{\text{ respectively.}}}" loading="lazy"></dd></dl>

More recently, Dusart has proved<a href="#cite_note-33">[32]</a>

(Theorem 5.1) that

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}\right)\leq \pi (x)\leq {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}+{\frac {7.59}{\log ^{3}x}}\right),}</annotation>

</semantics>

</math><img src="./0e3d474163a87030bdccd50ed52f99a0b0ca6e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.201ex; height:7.509ex;" alt="{\displaystyle {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}\right)\leq \pi (x)\leq {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}+{\frac {7.59}{\log ^{3}x}}\right),}" loading="lazy"></dd></dl>

for x ≥ 88789 and x > 1, respectively.

Going in the other direction, an approximation for the nth prime, pn, is

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<annotation encoding="application/x-tex">{\displaystyle p_{n}=n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}+O\left({\frac {(\log \log n)^{2}}{(\log n)^{2}}}\right)\right).}</annotation>

</semantics>

</math><img src="./153eef9a1d35499f310be52f271a37006dbf1fb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:68.068ex; height:7.509ex;" alt="{\displaystyle p_{n}=n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}+O\left({\frac {(\log \log n)^{2}}{(\log n)^{2}}}\right)\right).}" loading="lazy"></dd></dl>

Here are some inequalities for the nth prime. The lower bound is due to Dusart (1999)<a href="#cite_note-34">[33]</a> and the upper bound to Rosser (1941).<a href="#cite_note-35">[34]</a>

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<annotation encoding="application/x-tex">{\displaystyle n(\log n+\log \log n-1)<p_{n}<n(\log n+\log \log n)\quad {\text{for }}n\geq 6.}</annotation>

</semantics>

</math><img src="./396f810308234bf1b5c825537cb1176b2abcb023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.4ex; height:2.843ex;" alt="{\displaystyle n(\log n+\log \log n-1)<p_{n}<n(\log n+\log \log n)\quad {\text{for }}n\geq 6.}" loading="lazy"></dd></dl>

The left inequality holds for n ≥ 2 and the right inequality holds for n ≥ 6. A variant form sometimes seen substitutes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log n+\log \log n=\log(n\log n).}">

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<annotation encoding="application/x-tex">{\displaystyle \log n+\log \log n=\log(n\log n).}</annotation>

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</math><img src="./c27e6d741232fc9573cbedaf06d996b0fedae1e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.768ex; height:2.843ex;" alt="{\displaystyle \log n+\log \log n=\log(n\log n).}" loading="lazy"> An even simpler lower bound is<a href="#cite_note-Rosser62-36">[35]</a>

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<annotation encoding="application/x-tex">{\displaystyle n\log n<p_{n},}</annotation>

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</math><img src="./a282580718f4cfbfee14f79e3bad276a36d9c165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.669ex; height:2.509ex;" alt="{\displaystyle n\log n<p_{n},}" loading="lazy"></dd></dl>

which holds for all n ≥ 1, but the lower bound above is tighter for n > ee ≈15.154.

In 2010 Dusart proved<a href="#cite_note-Dusart2010-32">[31]</a> (Propositions 6.7 and 6.6) that

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<annotation encoding="application/x-tex">{\displaystyle n\left(\log n+\log \log n-1+{\frac {\log \log n-2.1}{\log n}}\right)\leq p_{n}\leq n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}\right),}</annotation>

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</math><img src="./79b1435711c469c72fc77c70dcef047d2c340622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:92.836ex; height:6.176ex;" alt="{\displaystyle n\left(\log n+\log \log n-1+{\frac {\log \log n-2.1}{\log n}}\right)\leq p_{n}\leq n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}\right),}" loading="lazy"></dd></dl>

for n ≥ 3 and n ≥ 688383, respectively.

In 2024, Axler<a href="#cite_note-37">[36]</a> further tightened this (equations 1.12 and 1.13) using bounds of the form

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<annotation encoding="application/x-tex">{\displaystyle f(n,g(w))=n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}-{\frac {g(\log \log n)}{2\log ^{2}n}}\right)}</annotation>

</semantics>

</math><img src="./ad991566746a5470e6a419d034c2dafc479b145c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.883ex; height:7.509ex;" alt="{\displaystyle f(n,g(w))=n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}-{\frac {g(\log \log n)}{2\log ^{2}n}}\right)}" loading="lazy"></dd></dl>

proving that

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n,w^{2}-6w+11.321)\leq p_{n}\leq f(n,w^{2}-6w)}</annotation>

</semantics>

</math><img src="./4455ff030ce40cb499d6244f0c44f90e61394aa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.688ex; height:3.176ex;" alt="{\displaystyle f(n,w^{2}-6w+11.321)\leq p_{n}\leq f(n,w^{2}-6w)}" loading="lazy"></dd></dl>

for n ≥ 2 and n ≥ 3468, respectively.

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

There are additional bounds of varying complexity.<a href="#cite_note-38">[37]</a><a href="#cite_note-39">[38]</a><a href="#cite_note-40">[39]</a>

<div class="mw-heading mw-heading2"><h2 id="The_Riemann_hypothesis">The Riemann hypothesis</h2></div>

The <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> implies a much tighter bound on the error in the estimate for π(x), and hence to a more regular distribution of prime numbers,

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (x)=\operatorname {li} (x)+O({\sqrt {x}}\log {x}).}</annotation>

</semantics>

</math><img src="./d3759e9342f7faedea3c56308dea788ea80d3f73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.413ex; height:3.009ex;" alt="{\displaystyle \pi (x)=\operatorname {li} (x)+O({\sqrt {x}}\log {x}).}" loading="lazy"></dd></dl>

Specifically,<a href="#cite_note-41">[40]</a>

</mrow>

</mrow>

<mfrac>

<msqrt>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {\sqrt {x}}{8\pi }}\,\log {x},\quad {\text{for all }}x\geq 2657.}</annotation>

</semantics>

</math><img src="./13655191a0d5b1e7e78e7cfda6d68e6362f7356b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.185ex; height:5.843ex;" alt="{\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {\sqrt {x}}{8\pi }}\,\log {x},\quad {\text{for all }}x\geq 2657.}" loading="lazy"></dd></dl>

<a href="#CITEREFDudek2015">Dudek (2015)</a> proved that the Riemann hypothesis implies that for all x ≥ 2 there is a prime p satisfying

<mfrac>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x.}</annotation>

</semantics>

</math><img src="./584244aaeca942988ad08219c8b2807f63b0b7b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.022ex; height:5.176ex;" alt="{\displaystyle x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x.}" loading="lazy"></dd></dl>

<ul><li><a href="Bertrand's_postulate" title="Bertrand's postulate">Bertrand's postulate</a></li>

<li><a href="Oppermann's_conjecture" title="Oppermann's conjecture">Oppermann's conjecture</a></li>

<li><a href="Ramanujan_prime" title="Ramanujan prime">Ramanujan prime</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Dusart2010-32">^ <a href="#cite_ref-Dusart2010_32-0">a</a> <a href="#cite_ref-Dusart2010_32-1">b</a> <cite id="CITEREFDusart2010" class="citation arxiv cs1"><a href="Pierre_Dusart" title="Pierre Dusart">Dusart, Pierre</a> (2 Feb 2010). "Estimates of Some Functions Over Primes without R.H.". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1002.0442v1">1002.0442v1</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.NT">math.NT</a>].</cite>

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<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFDusart1999" class="citation journal cs1"><a href="Pierre_Dusart" title="Pierre Dusart">Dusart, Pierre</a> (January 1999). <a rel="nofollow" class="external text" href="https://www.ams.org/mcom/1999-68-225/S0025-5718-99-01037-6/S0025-5718-99-01037-6.pdf">"The kth prime is greater than k(ln k + ln ln k − 1) for k ≥ 2"</a> (PDF). Mathematics of Computation. 68 (225): 411–415. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999MaCom..68..411D">1999MaCom..68..411D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-99-01037-6">10.1090/S0025-5718-99-01037-6</a>.</cite>

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<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite id="CITEREFRosser1941" class="citation journal cs1"><a href="J._Barkley_Rosser" title="J. Barkley Rosser">Rosser, Barkley</a> (January 1941). "Explicit bounds for some functions of prime numbers". American Journal of Mathematics. 63 (1): 211–232. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371291">10.2307/2371291</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371291">2371291</a>.</cite>

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<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFAxler2024" class="citation journal cs1">Axler, Christian (2024) [11 Mar 2022]. <a rel="nofollow" class="external text" href="https://math.colgate.edu/~integers/y34/y34.pdf">"Effective Estimates for Some Functions Defined over Primes"</a> (PDF). Integers. 24 A34. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2203.05917">2203.05917</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5281%2Fzenodo.10677755">10.5281/zenodo.10677755</a>.</cite>

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<li id="cite_note-41"><a href="#cite_ref-41">^</a> <cite id="CITEREFSchoenfeld1976" class="citation journal cs1"><a href="Lowell_Schoenfeld" title="Lowell Schoenfeld">Schoenfeld, Lowell</a> (1976). "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II". <a href="Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a>. 30 (134). American Mathematical Society: 337–360. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2005976">10.2307/2005976</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5718">0025-5718</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2005976">2005976</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0457374">0457374</a>.</cite>

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</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="Hugh_Lowell_Montgomery" title="Hugh Lowell Montgomery">Montgomery</a> showed that (assuming the Riemann hypothesis) at least two thirds of all zeros are simple.

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Chris Caldwell, <a rel="nofollow" class="external text" href="http://primes.utm.edu/nthprime/">The Nth Prime Page</a> at The <a href="Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a>.</li>

<li>Tomás Oliveira e Silva, <a rel="nofollow" class="external text" href="http://sweet.ua.pt/tos/primes.html">Tables of prime-counting functions</a>.</li>

<li><cite id="CITEREFDudek2015" class="citation cs2">Dudek, Adrian W. (2015), "On the Riemann hypothesis and the difference between primes", International Journal of Number Theory, 11 (3): 771–778, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1402.6417">1402.6417</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014arXiv1402.6417D">2014arXiv1402.6417D</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS1793042115500426">10.1142/S1793042115500426</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1793-0421">1793-0421</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119321107">119321107</a></cite></li></ul></div><div><div class="zim-footer">

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