Prime-counting function (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Prime-counting function" in English language version.

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13doi.org

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7arxiv.org

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3mersenneforum.org

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2projecteuclid.org

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1harvard.edu

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ams.org

Riesel, Hans; Göhl, Gunnar (1970). "Some calculations related to Riemann's prime number formula" (PDF). Mathematics of Computation. 24 (112). American Mathematical Society: 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR 2004630. MR 0277489.
Deléglise, Marc; Rivat, Joel (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.
Dusart, Pierre (January 1999). "The k^th prime is greater than k(ln k + ln ln k − 1) for k ≥ 2" (PDF). Mathematics of Computation. 68 (225): 411–415. Bibcode:1999MaCom..68..411D. doi:10.1090/S0025-5718-99-01037-6.

mathscinet.ams.org

Riesel, Hans; Göhl, Gunnar (1970). "Some calculations related to Riemann's prime number formula" (PDF). Mathematics of Computation. 24 (112). American Mathematical Society: 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR 2004630. MR 0277489.
Schoenfeld, Lowell (1976). "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II". Mathematics of Computation. 30 (134). American Mathematical Society: 337–360. doi:10.2307/2005976. ISSN 0025-5718. JSTOR 2005976. MR 0457374.

arxiv.org

Kevin Ford (November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function" (PDF). Proc. London Math. Soc. 85 (3): 565–633. arXiv:1910.08209. doi:10.1112/S0024611502013655. S2CID 121144007.
Mossinghoff, Michael J.; Trudgian, Timothy S. (2015). "Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function". J. Number Theory. 157: 329–349. arXiv:1410.3926. doi:10.1016/J.JNT.2015.05.010. S2CID 117968965.
Platt, David J. (May 2015) [March 2012]. "Computing $π (x)$ Analytically". Mathematics of Computation. 84 (293): 1521–1535. arXiv:1203.5712. doi:10.1090/S0025-5718-2014-02884-6.
Dusart, Pierre (2 Feb 2010). "Estimates of Some Functions Over Primes without R.H.". arXiv:1002.0442v1 [math.NT].
Axler, Christian (2019) [23 Mar 2017]. "New estimates for the nth prime number". Journal of Integer Sequences. 19 (4) 2. arXiv:1706.03651.
Axler, Christian (2018) [23 Mar 2017]. "New Estimates for Some Functions Defined Over Primes" (PDF). Integers. 18 A52. arXiv:1703.08032. doi:10.5281/zenodo.10677755.
Axler, Christian (2024) [11 Mar 2022]. "Effective Estimates for Some Functions Defined over Primes" (PDF). Integers. 24 A34. arXiv:2203.05917. doi:10.5281/zenodo.10677755.

books.google.com

Berndt, Bruce C. (2012-12-06). Ramanujan's Notebooks, Part IV. Springer Science & Business Media. pp. 112–113. ISBN 9781461269328.

colgate.edu

math.colgate.edu

Axler, Christian (2018) [23 Mar 2017]. "New Estimates for Some Functions Defined Over Primes" (PDF). Integers. 18 A52. arXiv:1703.08032. doi:10.5281/zenodo.10677755.
Axler, Christian (2024) [11 Mar 2022]. "Effective Estimates for Some Functions Defined over Primes" (PDF). Integers. 24 A34. arXiv:2203.05917. doi:10.5281/zenodo.10677755.

dal.ca

dalspace.library.dal.ca

Staple, Douglas (19 August 2015). The combinatorial algorithm for computing π(x) (Thesis). Dalhousie University. Retrieved 2015-09-01.

doi.org

Kevin Ford (November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function" (PDF). Proc. London Math. Soc. 85 (3): 565–633. arXiv:1910.08209. doi:10.1112/S0024611502013655. S2CID 121144007.
Mossinghoff, Michael J.; Trudgian, Timothy S. (2015). "Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function". J. Number Theory. 157: 329–349. arXiv:1410.3926. doi:10.1016/J.JNT.2015.05.010. S2CID 117968965.
Riesel, Hans; Göhl, Gunnar (1970). "Some calculations related to Riemann's prime number formula" (PDF). Mathematics of Computation. 24 (112). American Mathematical Society: 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR 2004630. MR 0277489.
Platt, David J. (May 2015) [March 2012]. "Computing $π (x)$ Analytically". Mathematics of Computation. 84 (293): 1521–1535. arXiv:1203.5712. doi:10.1090/S0025-5718-2014-02884-6.
Deléglise, Marc; Rivat, Joel (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.
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.
Dusart, Pierre (January 2018). "Explicit estimates of some functions over primes". Ramanujan Journal. 45 (1): 225–234. doi:10.1007/s11139-016-9839-4. S2CID 125120533.
Dusart, Pierre (January 1999). "The k^th prime is greater than k(ln k + ln ln k − 1) for k ≥ 2" (PDF). Mathematics of Computation. 68 (225): 411–415. Bibcode:1999MaCom..68..411D. doi:10.1090/S0025-5718-99-01037-6.
Rosser, Barkley (January 1941). "Explicit bounds for some functions of prime numbers". American Journal of Mathematics. 63 (1): 211–232. doi:10.2307/2371291. JSTOR 2371291.
Rosser, J. Barkley; Schoenfeld, Lowell (March 1962). "Approximate formulas for some functions of prime numbers". Illinois Journal of Mathematics. 6 (1): 64–94. doi:10.1215/ijm/1255631807.
Axler, Christian (2018) [23 Mar 2017]. "New Estimates for Some Functions Defined Over Primes" (PDF). Integers. 18 A52. arXiv:1703.08032. doi:10.5281/zenodo.10677755.
Axler, Christian (2024) [11 Mar 2022]. "Effective Estimates for Some Functions Defined over Primes" (PDF). Integers. 24 A34. arXiv:2203.05917. doi:10.5281/zenodo.10677755.
Schoenfeld, Lowell (1976). "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II". Mathematics of Computation. 30 (134). American Mathematical Society: 337–360. doi:10.2307/2005976. ISSN 0025-5718. JSTOR 2005976. MR 0457374.

ex.ac.uk

secamlocal.ex.ac.uk

"The encoding of the prime distribution by the zeta zeros". Matthew Watkins. Retrieved 2008-09-14.

free.fr

numbers.computation.free.fr

"A table of values of $π (x)$ ". Xavier Gourdon, Pascal Sebah, Patrick Demichel. Retrieved 2008-09-14.

harvard.edu

ui.adsabs.harvard.edu

Dusart, Pierre (January 1999). "The k^th prime is greater than k(ln k + ln ln k − 1) for k ≥ 2" (PDF). Mathematics of Computation. 68 (225): 411–415. Bibcode:1999MaCom..68..411D. doi:10.1090/S0025-5718-99-01037-6.

illinois.edu

faculty.math.illinois.edu

Kevin Ford (November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function" (PDF). Proc. London Math. Soc. 85 (3): 565–633. arXiv:1910.08209. doi:10.1112/S0024611502013655. S2CID 121144007.

jstor.org

Riesel, Hans; Göhl, Gunnar (1970). "Some calculations related to Riemann's prime number formula" (PDF). Mathematics of Computation. 24 (112). American Mathematical Society: 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR 2004630. MR 0277489.
Rosser, Barkley (January 1941). "Explicit bounds for some functions of prime numbers". American Journal of Mathematics. 63 (1): 211–232. doi:10.2307/2371291. JSTOR 2371291.
Schoenfeld, Lowell (1976). "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II". Mathematics of Computation. 30 (134). American Mathematical Society: 337–360. doi:10.2307/2005976. ISSN 0025-5718. JSTOR 2005976. MR 0457374.

mersenneforum.org

Walisch, Kim (September 6, 2015). "New confirmed π(10²⁷) prime counting function record". Mersenne Forum.
Baugh, David (August 30, 2020). "New prime counting function record, pi(10^28)". Mersenne Forum.
Walisch, Kim (March 4, 2022). "New prime counting function record: PrimePi(10^29)". Mersenne Forum.

projecteuclid.org

Lehmer, Derrick Henry (1 April 1958). "On the exact number of primes less than a given limit". Illinois J. Math. 3 (3): 381–388. Retrieved 1 February 2017.
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.

semanticscholar.org

api.semanticscholar.org

Kevin Ford (November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function" (PDF). Proc. London Math. Soc. 85 (3): 565–633. arXiv:1910.08209. doi:10.1112/S0024611502013655. S2CID 121144007.
Mossinghoff, Michael J.; Trudgian, Timothy S. (2015). "Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function". J. Number Theory. 157: 329–349. arXiv:1410.3926. doi:10.1016/J.JNT.2015.05.010. S2CID 117968965.
Dusart, Pierre (January 2018). "Explicit estimates of some functions over primes". Ramanujan Journal. 45 (1): 225–234. doi:10.1007/s11139-016-9839-4. S2CID 125120533.

stackexchange.com

math.stackexchange.com

"Bounds for n-th prime". Mathematics StackExchange. 31 December 2015.

t5k.org

Franke, Jens (2010-07-29). "Conditional Calculation of π(10²⁴)". Chris K. Caldwell. Retrieved 2024-03-30.

tum.de

www-m3.ma.tum.de

Bornemann, Folkmar. "Solution of a Problem Posed by Jörg Waldvogel" (PDF).

ua.pt

sweet.ua.pt

"Tables of values of $π (x)$ and of $π 2 (x)$ ". Tomás Oliveira e Silva. Retrieved 2024-03-31.

uni-bonn.de

math.uni-bonn.de

"Analytic Computation of the prime-counting Function". J. Buethe. 27 May 2014. Retrieved 2015-09-01. Includes 600,000 value of $π (x)$ for $1014 \leq x \leq 1.6\times10 18$

uqam.ca

ism.uqam.ca

Hutama, Daniel (2017). "Implementation of Riemann's Explicit Formula for Rational and Gaussian Primes in Sage" (PDF). Institut des sciences mathématiques.

utm.edu

primes.utm.edu

"How many primes are there?". Chris K. Caldwell. Archived from the original on 2012-10-15. Retrieved 2008-12-02.

uwaterloo.ca

cs.uwaterloo.ca

Axler, Christian (2019) [23 Mar 2017]. "New estimates for the nth prime number". Journal of Integer Sequences. 19 (4) 2. arXiv:1706.03651.

web.archive.org

"How many primes are there?". Chris K. Caldwell. Archived from the original on 2012-10-15. Retrieved 2008-12-02.

wolfram.com

mathworld.wolfram.com

worldcat.org

search.worldcat.org

Riesel, Hans; Göhl, Gunnar (1970). "Some calculations related to Riemann's prime number formula" (PDF). Mathematics of Computation. 24 (112). American Mathematical Society: 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR 2004630. MR 0277489.
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.
Schoenfeld, Lowell (1976). "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II". Mathematics of Computation. 30 (134). American Mathematical Society: 337–360. doi:10.2307/2005976. ISSN 0025-5718. JSTOR 2005976. MR 0457374.

zbmath.org

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.