Prime-counting function

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

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

History

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

${\displaystyle {\frac {x}{\ln(x)}}}$

in the sense that

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

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

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

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]

In 1899, de la Vallée Poussin proved that (see also Theorem 23 of[6])

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

for some positive constant a. Here, O(...) is the big O notation.

More precise estimates of ${\displaystyle \pi (x)\!}$ are now known. For example, in 2002, Kevin Ford proved that[7]

${\displaystyle \pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-0.2098(\ln x)^{\frac {3}{5}}(\ln \ln x)^{-{\frac {1}{5}}}\right)\right)}$.

In 2016, Tim Trudgian proved an explicit upper bound for the difference between ${\displaystyle \pi (x)}$ and ${\displaystyle \operatorname {li} (x)}$:

${\displaystyle {\big |}\pi (x)-\operatorname {li} (x){\big |}\leq 0.2795{\frac {x}{(\ln x)^{3/4}}}\exp \left(-{\sqrt {\frac {\ln x}{6.455}}}\right)}$

for ${\displaystyle x\geq 229}$.[8]

For most values of ${\displaystyle x}$ we are interested in (i.e., when ${\displaystyle x}$ is not unreasonably large) ${\displaystyle \operatorname {li} (x)\!}$ is greater than ${\displaystyle \pi (x)\!}$. However, ${\displaystyle \pi (x)-\operatorname {li} (x)}$ is known to change sign infinitely many times. For a discussion of this, see Skewes' number.

Exact form

Of profound importance, Bernhard Riemann proved that the prime-counting function is exactly[9]

${\displaystyle \pi (x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })}$

where

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

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 ln x). Equivalently, 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(x) may be written

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

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

Table of π(x), x / ln x, and li(x)

The table shows how the three functions π(x), x / ln x and li(x) compare at powers of 10. See also,[3][10][11] and[12]

x	π(x)	π(x) − x / ln x	li(x) − π(x)	x / π(x)	x / ln x % Error
10	4	−0.3	2.2	2.500	-7.5%
10^2	25	3.3	5.1	4.000	13.20%
10^3	168	23	10	5.952	13.69%
10^4	1,229	143	17	8.137	11.64%
10^5	9,592	906	38	10.425	9.45%
10^6	78,498	6,116	130	12.740	7.79%
10^7	664,579	44,158	339	15.047	6.64%
10^8	5,761,455	332,774	754	17.357	5.78%
10^9	50,847,534	2,592,592	1,701	19.667	5.10%
10^10	455,052,511	20,758,029	3,104	21.975	4.56%
10^11	4,118,054,813	169,923,159	11,588	24.283	4.13%
10^12	37,607,912,018	1,416,705,193	38,263	26.590	3.77%
10^13	346,065,536,839	11,992,858,452	108,971	28.896	3.47%
10^14	3,204,941,750,802	102,838,308,636	314,890	31.202	3.21%
10^15	29,844,570,422,669	891,604,962,452	1,052,619	33.507	2.99%
10^16	279,238,341,033,925	7,804,289,844,393	3,214,632	35.812	2.79%
10^17	2,623,557,157,654,233	68,883,734,693,281	7,956,589	38.116	2.63%
10^18	24,739,954,287,740,860	612,483,070,893,536	21,949,555	40.420	2.48%
10^19	234,057,667,276,344,607	5,481,624,169,369,960	99,877,775	42.725	2.34%
10^20	2,220,819,602,560,918,840	49,347,193,044,659,701	222,744,644	45.028	2.22%
10^21	21,127,269,486,018,731,928	446,579,871,578,168,707	597,394,254	47.332	2.11%
10^22	201,467,286,689,315,906,290	4,060,704,006,019,620,994	1,932,355,208	49.636	2.02%
10^23	1,925,320,391,606,803,968,923	37,083,513,766,578,631,309	7,250,186,216	51.939	1.93%
10^24	18,435,599,767,349,200,867,866	339,996,354,713,708,049,069	17,146,907,278	54.243	1.84%
10^25	176,846,309,399,143,769,411,680	3,128,516,637,843,038,351,228	55,160,980,939	56.546	1.77%
10^26	1,699,246,750,872,437,141,327,603	28,883,358,936,853,188,823,261	155,891,678,121	58.850	1.70%
10^27	16,352,460,426,841,680,446,427,399	267,479,615,610,131,274,163,365	508,666,658,006	61.153	1.64%

Graph showing ratio of the prime-counting function π(x) to two of its approximations, x/ln x and Li(x). As x increases (note x axis is logarithmic), both ratios tend towards 1. The ratio for x/ln x converges from above very slowly, while the ratio for Li(x) converges more quickly from below.

In the On-Line Encyclopedia of Integer Sequences, the π(x) column is sequence OEIS: A006880, π(x) − x/ln x is sequence OEIS: A057835, and li(x) − π(x) is sequence OEIS: A057752.

The value for π(10²⁴) was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.[13] It was later verified unconditionally in a computation by D. J. Platt.[14] The value for π(10²⁵) is due to J. Buethe, J. Franke, A. Jost, and T. Kleinjung.[15] The value for π(10²⁶) was computed by D. B. Staple.[16] All other prior entries in this table were also verified as part of that work.

The value for 10²⁷ was published in 2015 by David Baugh and Kim Walisch.[17]

Algorithms for evaluating π(x)

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

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

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

(where ${\displaystyle \lfloor {x}\rfloor }$ denotes the floor function). This number is therefore equal to

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

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

The Meissel–Lehmer algorithm

Main article: Meissel–Lehmer algorithm

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

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

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

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

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

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

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

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

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

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

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

where the sum is over prime numbers.

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

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

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

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise and Rivat.[18]

Other prime-counting functions

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

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

where p is a prime.

We may also write

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

where Λ(n) is the von Mangoldt function and

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

The Möbius inversion formula then gives

${\displaystyle \pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi _{0}(x^{1/n})}$

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

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

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

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

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

Formulas for prime-counting functions

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 formulas.[19]

We have the following expression for ψ:

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

where

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

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

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

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

Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function

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

Thus, Möbius inversion formula gives us[20]

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

valid for x > 1, where

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

is Riemann's R-function[21] and μ(n) is the Möbius function. The latter series for it is known as Gram series[22][23]. Because ${\displaystyle \ln(x)<x}$ for all ${\displaystyle x>0}$, this series converges for all positive x by comparison with the series for ${\displaystyle e^{x}}$.

Δ-function (red line) on log scale

The sum over non-trivial zeta zeros in the formula for ${\displaystyle \pi _{0}(x)}$ describes the fluctuations of ${\displaystyle \pi _{0}(x),}$ while the remaining terms give the "smooth" part of prime-counting function,[24] so one can use

${\displaystyle \operatorname {R} (x)-{\frac {1}{\ln x}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}}$

as the best estimator of ${\displaystyle \pi (x)}$ for x > 1.

The amplitude of the "noisy" part is heuristically about ${\displaystyle {\sqrt {x}}/\ln x,}$ so the fluctuations of the distribution of primes may be clearly represented with the Δ-function:

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

An extensive table of the values of Δ(x) is available.[11]

Inequalities

Here are some useful inequalities for π(x).

${\displaystyle {\frac {x}{\ln x}}<\pi (x)<1.25506{\frac {x}{\ln x}}}$

for x ≥ 17.

The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is ${\displaystyle {\frac {30\ln 113}{113}}}$ to 5 decimal places, as ${\displaystyle {\frac {\pi (x)\ln x}{x}}}$ has its maximum value at x = 113.[25]

Pierre Dusart proved in 2010:

${\displaystyle {\frac {x}{\ln x-1}}<\pi (x)}$ for ${\displaystyle x\geq 5393}$, and

${\displaystyle \pi (x)<{\frac {x}{\ln x-1.1}}}$ for ${\displaystyle x\geq 60184}$.[26]

Here are some inequalities for the nth prime, p_n. The upper bound is due to Rosser (1941),[27] the lower one to Dusart (1999):[28]

${\displaystyle n(\ln(n\ln n)-1)<p_{n}<n{\ln(n\ln n)}\!}$ for n ≥ 6.

The left inequality holds for n ≥ 2 and the right inequality holds for n ≥ 6.

An approximation for the nth prime number is

${\displaystyle p_{n}=n(\ln(n\ln n)-1)+{\frac {n(\ln \ln n-2)}{\ln n}}+O\left({\frac {n(\ln \ln n)^{2}}{(\ln n)^{2}}}\right).}$

Ramanujan[29] proved that the inequality

${\displaystyle \pi (x)^{2}<{\frac {ex}{\log x}}\pi {\bigg (}{\frac {x}{e}}{\bigg )}}$

holds for all sufficiently large values of ${\displaystyle x}$.

In [26], Dusart proved (Proposition 6.6) that, for ${\displaystyle n\geq 688383}$,

${\displaystyle p_{n}\leq n\left(\ln n+\ln \ln n-1+{\frac {\ln \ln n-2}{\ln n}}\right)}$ ,

and (Proposition 6.7) that, for ${\displaystyle n\geq 3}$,

${\displaystyle p_{n}\geq n\left(\ln n+\ln \ln n-1+{\frac {\ln \ln n-2.1}{\ln n}}\right)}$ .

More recently, Dusart[30] has proved (Theorem 5.1) that, for ${\displaystyle x>1}$,

${\displaystyle \pi (x)\leq {\frac {x}{\ln x}}\left(1+{\frac {1}{\ln x}}+{\frac {2}{\ln ^{2}x}}+{\frac {7.59}{\ln ^{3}x}}\right)}$ ,

and that, for ${\displaystyle x\geq 88789}$,

${\displaystyle \pi (x)>{\frac {x}{\ln x}}\left(1+{\frac {1}{\ln x}}+{\frac {2}{\ln ^{2}x}}\right)}$ .

The Riemann hypothesis

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

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

Specifically,[31]

${\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {1}{8\pi }}{\sqrt {x}}\,\log {x},\qquad {\text{for all }}x\geq 2657.}$

See also

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

References

1. Bach, Eric; Shallit, Jeffrey (1996). Algorithmic Number Theory. MIT Press. volume 1 page 234 section 8.8. ISBN 0-262-02405-5.
2. Weisstein, Eric W. "Prime Counting Function". MathWorld.
3. "How many primes are there?". Chris K. Caldwell. Retrieved 2008-12-02.
4. Dickson, Leonard Eugene (2005). History of the Theory of Numbers, Vol. I: Divisibility and Primality. Dover Publications. ISBN 0-486-44232-2.
5. Ireland, Kenneth; Rosen, Michael (1998). A Classical Introduction to Modern Number Theory (Second ed.). Springer. ISBN 0-387-97329-X.
6. A. E. Ingham (2000). The Distribution of Prime Numbers. Cambridge University Press. ISBN 0-521-39789-8.
7. Kevin Ford (November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function" (PDF). Proc. London Math. Soc. 85 (3): 565–633. doi:10.1112/S0024611502013655.
8. Tim Trudgian (February 2016). "Updating the error term in the prime number theorem". Ramanujan Journal. 39 (2): 225–234. arXiv:1401.2689. doi:10.1007/s11139-014-9656-6.
9. "The Fluctuations of the Prime-counting Function pi(x)". www.primefan.ru. Retrieved 17 May 2019.
10. "Tables of values of pi(x) and of pi2(x)". Tomás Oliveira e Silva. Retrieved 2008-09-14.
11. "Values of π(x) and Δ(x) for various values of x". Andrey V. Kulsha. Retrieved 2008-09-14.
12. "A table of values of pi(x)". Xavier Gourdon, Pascal Sebah, Patrick Demichel. Retrieved 2008-09-14.
13. "Conditional Calculation of pi(10²⁴)". Chris K. Caldwell. Retrieved 2010-08-03.
14. Platt, David J. (2012). "Computing π(x) Analytically)". arXiv:1203.5712 [math.NT].
15. "How Many Primes Are There?". J. Buethe. Retrieved 2015-09-01.
16. Staple, Douglas (19 August 2015). The combinatorial algorithm for computing pi(x) (Thesis). Dalhousie University. Retrieved 2015-09-01.
17. Walisch, Kim (September 6, 2015). "New confirmed pi(10^27) prime counting function record". Mersenne Forum.
18. Marc Deleglise; Joel Rivat (January 1996). "Computing π(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko method" (PDF). Mathematics of Computation. 65 (213): 235–245. doi:10.1090/S0025-5718-96-00674-6.
19. Titchmarsh, E.C. (1960). The Theory of Functions, 2nd ed. Oxford University Press.
20. Riesel, Hans; Göhl, Gunnar (1970). "Some calculations related to Riemann's prime number formula". Mathematics of Computation. American Mathematical Society. 24 (112): 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR 2004630. MR 0277489.
21. Weisstein, Eric W. "Riemann Prime Counting Function". MathWorld.
22. Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. 126 (2nd ed.). Birkhäuser. pp. 50–51. ISBN 0-8176-3743-5.
23. Weisstein, Eric W. "Gram Series". MathWorld.
24. "The encoding of the prime distribution by the zeta zeros". Matthew Watkins. Retrieved 2008-09-14.
25. Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers". Illinois J. Math. 6: 64–94. doi:10.1215/ijm/1255631807. ISSN 0019-2082. Zbl 0122.05001.
26. Dusart, Pierre (2 Feb 2010). "Estimates of Some Functions Over Primes without R.H.". arXiv:1002.0442v1 [math.NT].
27. Rosser, Barkley (1941). "Explicit bounds for some functions of prime numbers". American Journal of Mathematics. 63 (1): 211–232. doi:10.2307/2371291. JSTOR 2371291.
28. Dusart, Pierre (1999). "The kth prime is greater than k(lnk+lnlnk-1) for k>=2". Mathematics of Computation. 68 (225): 411–415. doi:10.1090/S0025-5718-99-01037-6.
29. Berndt, Bruce C. (2012-12-06). Ramanujan's Notebooks, Part IV. Springer Science & Business Media. pp. 112–113. ISBN 9781461269328.
30. Dusart, Pierre (January 2018). "Explicit estimates of some functions over primes". Ramanujan Journal. 45 (1): 225–234. doi:10.1007/s11139-016-9839-4.
31. Schoenfeld, Lowell (1976). "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II". Mathematics of Computation. American Mathematical Society. 30 (134): 337–360. doi:10.2307/2005976. ISSN 0025-5718. JSTOR 2005976. MR 0457374.

External links

• Chris Caldwell, The Nth Prime Page at The Prime Pages.
• Tomás Oliveira e Silva, Tables of prime-counting functions.