Prime number theorem (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Prime number theorem" in English language version.

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25doi.org

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18ams.org

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10jstor.org

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10semanticscholar.org

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10arxiv.org

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4nih.gov

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3zbmath.org

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2archive.org

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2harvard.edu

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1columbia.edu

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1terrytao.wordpress.com

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1illinois.edu

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amisbnf.org (Global: low place; English: low place)

sciences.amisbnf.org

de la Vallée Poussin, Charles-Jean (1896), "Recherches analytiques sur la théorie des nombres premiers.", Annales de la Société scientifique de Bruxelles, 20 B, 21 B, Imprimeur de l'Académie Royale de Belgique: 183–256, 281–352, 363–397, 351–368

ams.org (Global: 451^st place; English: 277^th place)

mathscinet.ams.org

Hoffman, Paul (1998). The Man Who Loved Only Numbers. New York: Hyperion Books. p. 227. ISBN 978-0-7868-8406-3. MR 1666054.
Goldfeld, Dorian (2004). "The elementary proof of the prime number theorem: an historical perspective" (PDF). In Chudnovsky, David; Chudnovsky, Gregory; Nathanson, Melvyn (eds.). Number theory (New York, 2003). New York: Springer-Verlag. pp. 179–192. doi:10.1007/978-1-4419-9060-0_10. ISBN 978-0-387-40655-8. MR 2044518.
Selberg, Atle (1949), "An Elementary Proof of the Prime-Number Theorem", Annals of Mathematics, 50 (2): 305–313, doi:10.2307/1969455, JSTOR 1969455, MR 0029410, S2CID 124153092
Newman, D.J. (1980). "Simple analytic proof of the prime number theorem". American Mathematical Monthly. 87 (9): 693–696. doi:10.2307/2321853. JSTOR 2321853. MR 0602825.
Zagier, Don (1997). "Newman's short proof of the prime number theorem". American Mathematical Monthly. 104 (8): 705–708. doi:10.2307/2975232. JSTOR 2975232. MR 1476753 – via maa.org.
von Koch, Helge (1901). "Sur la distribution des nombres premiers" [On the distribution of prime numbers]. Acta Mathematica (in French). 24 (1): 159–182. doi:10.1007/BF02403071. MR 1554926. S2CID 119914826.
Schoenfeld, Lowell (1976). "Sharper Bounds for the Chebyshev Functions $ϑ (x)$ and $ψ (x)$ . II". Mathematics of Computation. 30 (134): 337–360. doi:10.2307/2005976. JSTOR 2005976. MR 0457374.
Jørgensen, Bent; Martínez, José Raúl; Tsao, Min (1994). "Asymptotic behaviour of the variance function". Scandinavian Journal of Statistics. 21 (3): 223–243. JSTOR 4616314. MR 1292637.
Baas, Nils A.; Skau, Christian F. (2008). "The lord of the numbers, Atle Selberg. On his life and mathematics" (PDF). Bull. Amer. Math. Soc. 45 (4): 617–649. doi:10.1090/S0273-0979-08-01223-8. MR 2434348.
Cornaros, Charalambos; Dimitracopoulos, Costas (1994). "The prime number theorem and fragments of PA" (PDF). Archive for Mathematical Logic. 33 (4): 265–281. doi:10.1007/BF01270626. MR 1294272. S2CID 29171246. Archived from the original (PDF) on 2011-07-21.
Avigad, Jeremy; Donnelly, Kevin; Gray, David; Raff, Paul (2008). "A formally verified proof of the prime number theorem". ACM Transactions on Computational Logic. 9 (1): 2. arXiv:cs/0509025. doi:10.1145/1297658.1297660. MR 2371488. S2CID 7720253.
Harrison, John (2009). "Formalizing an analytic proof of the Prime Number Theorem". Journal of Automated Reasoning. 43 (3): 243–261. CiteSeerX 10.1.1.646.9725. doi:10.1007/s10817-009-9145-6. MR 2544285. S2CID 8032103.
Granville, Andrew; Martin, Greg (2006). "Prime number races" (PDF). American Mathematical Monthly. 113 (1): 1–33. doi:10.2307/27641834. JSTOR 27641834. MR 2202918.
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. MR 0003018.
Arias de Reyna, Juan; Toulisse, Jérémy (2013). "The n-th prime asymptotically". Journal de théorie des nombres de Bordeaux. 25 (3): 521–555. arXiv:1203.5413. doi:10.5802/jtnb.847. MR 3179675. Zbl 1298.11093.
Dusart, Pierre (1999). "The $k$ th prime is greater than $k (log k + log log k - 1)$ for $k \geq 2$ ". Mathematics of Computation. 68 (225): 411–415. doi:10.1090/S0025-5718-99-01037-6. MR 1620223.
Platt, David (2015). "Computing $π (x)$ analytically". Mathematics of Computation. 84 (293): 1521–1535. arXiv:1203.5712. doi:10.1090/S0025-5718-2014-02884-6. MR 3315519. S2CID 119174627.

ams.org

Baas, Nils A.; Skau, Christian F. (2008). "The lord of the numbers, Atle Selberg. On his life and mathematics" (PDF). Bull. Amer. Math. Soc. 45 (4): 617–649. doi:10.1090/S0273-0979-08-01223-8. MR 2434348.

archive.org (Global: 6^th place; English: 6^th place)

Hoffman, Paul (1998). The Man Who Loved Only Numbers. New York: Hyperion Books. p. 227. ISBN 978-0-7868-8406-3. MR 1666054.
Gauss, C. F. (1863), Werke, vol. 2 (1st ed.), Göttingen: Teubner, pp. 444–447.

arxiv.org (Global: 69^th place; English: 59^th place)

Kevin Ford (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.
Timothy 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. S2CID 11013503.
Avigad, Jeremy; Donnelly, Kevin; Gray, David; Raff, Paul (2008). "A formally verified proof of the prime number theorem". ACM Transactions on Computational Logic. 9 (1): 2. arXiv:cs/0509025. doi:10.1145/1297658.1297660. MR 2371488. S2CID 7720253.
Bennett, Michael A.; Martin, Greg; O'Bryant, Kevin; Rechnitzer, Andrew (2018). "Explicit bounds for primes in arithmetic progressions". Illinois J. Math. 62 (1–4): 427–532. arXiv:1802.00085. doi:10.1215/ijm/1552442669. S2CID 119647640.
Lemke Oliver, Robert J.; Soundararajan, Kannan (2016-08-02). "Unexpected biases in the distribution of consecutive primes". Proceedings of the National Academy of Sciences. 113 (31): E4446-54. arXiv:1603.03720. Bibcode:2016PNAS..113E4446L. doi:10.1073/pnas.1605366113. ISSN 0027-8424. PMC 4978288. PMID 27418603.
Dusart, Pierre (2 February 2010). "Estimates of some functions over primes, without R.H.". arXiv:1002.0442 [math.NT].
Arias de Reyna, Juan; Toulisse, Jérémy (2013). "The n-th prime asymptotically". Journal de théorie des nombres de Bordeaux. 25 (3): 521–555. arXiv:1203.5413. doi:10.5802/jtnb.847. MR 3179675. Zbl 1298.11093.
Axler, Christian (2019). "New Estimates for the $n$ th Prime Number". Journal of Integer Sequences. 22 19.4.2. arXiv:1706.03651.
Platt, David (2015). "Computing $π (x)$ analytically". Mathematics of Computation. 84 (293): 1521–1535. arXiv:1203.5712. doi:10.1090/S0025-5718-2014-02884-6. MR 3315519. S2CID 119174627.
Chebolu, Sunil; Mináč, Ján (December 2011). "Counting Irreducible Polynomials over Finite Fields Using the Inclusion $π$ Exclusion Principle". Mathematics Magazine. 84 (5): 369–371. arXiv:1001.0409. doi:10.4169/math.mag.84.5.369. JSTOR 10.4169/math.mag.84.5.369. S2CID 115181186.

bnf.fr (Global: 124^th place; English: 544^th place)

gallica.bnf.fr

Cesàro, Ernesto (1894). "Sur une formule empirique de M. Pervouchine". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (in French). 119: 848–849.

books.google.com (Global: 3^rd place; English: 3^rd place)

de la Vallée Poussin, Charles-Jean (1899), "Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs a une limite donnée.", Mémoires couronnés de l'Académie de Belgique, 59, Imprimeur de l'Académie Royale de Belgique: 1–74

cam.ac.uk (Global: 670^th place; English: 480^th place)

cl.cam.ac.uk

Harrison, John (2009). "Formalizing an analytic proof of the Prime Number Theorem". Journal of Automated Reasoning. 43 (3): 243–261. CiteSeerX 10.1.1.646.9725. doi:10.1007/s10817-009-9145-6. MR 2544285. S2CID 8032103.

columbia.edu (Global: 488^th place; English: 374^th place)

math.columbia.edu

Goldfeld, Dorian (2004). "The elementary proof of the prime number theorem: an historical perspective" (PDF). In Chudnovsky, David; Chudnovsky, Gregory; Nathanson, Melvyn (eds.). Number theory (New York, 2003). New York: Springer-Verlag. pp. 179–192. doi:10.1007/978-1-4419-9060-0_10. ISBN 978-0-387-40655-8. MR 2044518.

doi.org (Global: 2^nd place; English: 2^nd place)

Apostol, Tom M. (1976). Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics (1 ed.). Springer. doi:10.1007/978-1-4757-5579-4. ISBN 978-1-4757-5579-4.
Costa Pereira, N. (August–September 1985). "A short proof of Chebyshev's theorem". American Mathematical Monthly. 92 (7): 494–495. doi:10.2307/2322510. JSTOR 2322510.
Nair, M. (February 1982). "On Chebyshev-type inequalities for primes". American Mathematical Monthly. 89 (2): 126–129. doi:10.2307/2320934. JSTOR 2320934.
Goldfeld, Dorian (2004). "The elementary proof of the prime number theorem: an historical perspective" (PDF). In Chudnovsky, David; Chudnovsky, Gregory; Nathanson, Melvyn (eds.). Number theory (New York, 2003). New York: Springer-Verlag. pp. 179–192. doi:10.1007/978-1-4419-9060-0_10. ISBN 978-0-387-40655-8. MR 2044518.
Selberg, Atle (1949), "An Elementary Proof of the Prime-Number Theorem", Annals of Mathematics, 50 (2): 305–313, doi:10.2307/1969455, JSTOR 1969455, MR 0029410, S2CID 124153092
Erdős, Paul (July 1949). "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem" (PDF). Proceedings of the National Academy of Sciences of the U.S.A. 35 (7): 374–384. Bibcode:1949PNAS...35..374E. doi:10.1073/pnas.35.7.374. PMC 1063042. PMID 16588909 – via renyi.hu.
Newman, D.J. (1980). "Simple analytic proof of the prime number theorem". American Mathematical Monthly. 87 (9): 693–696. doi:10.2307/2321853. JSTOR 2321853. MR 0602825.
Zagier, Don (1997). "Newman's short proof of the prime number theorem". American Mathematical Monthly. 104 (8): 705–708. doi:10.2307/2975232. JSTOR 2975232. MR 1476753 – via maa.org.
Kevin Ford (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.
Timothy 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. S2CID 11013503.
von Koch, Helge (1901). "Sur la distribution des nombres premiers" [On the distribution of prime numbers]. Acta Mathematica (in French). 24 (1): 159–182. doi:10.1007/BF02403071. MR 1554926. S2CID 119914826.
Schoenfeld, Lowell (1976). "Sharper Bounds for the Chebyshev Functions $ϑ (x)$ and $ψ (x)$ . II". Mathematics of Computation. 30 (134): 337–360. doi:10.2307/2005976. JSTOR 2005976. MR 0457374.
Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes". Acta Mathematica. 41: 119–196. doi:10.1007/BF02422942.
Baas, Nils A.; Skau, Christian F. (2008). "The lord of the numbers, Atle Selberg. On his life and mathematics" (PDF). Bull. Amer. Math. Soc. 45 (4): 617–649. doi:10.1090/S0273-0979-08-01223-8. MR 2434348.
Cornaros, Charalambos; Dimitracopoulos, Costas (1994). "The prime number theorem and fragments of PA" (PDF). Archive for Mathematical Logic. 33 (4): 265–281. doi:10.1007/BF01270626. MR 1294272. S2CID 29171246. Archived from the original (PDF) on 2011-07-21.
Avigad, Jeremy; Donnelly, Kevin; Gray, David; Raff, Paul (2008). "A formally verified proof of the prime number theorem". ACM Transactions on Computational Logic. 9 (1): 2. arXiv:cs/0509025. doi:10.1145/1297658.1297660. MR 2371488. S2CID 7720253.
Harrison, John (2009). "Formalizing an analytic proof of the Prime Number Theorem". Journal of Automated Reasoning. 43 (3): 243–261. CiteSeerX 10.1.1.646.9725. doi:10.1007/s10817-009-9145-6. MR 2544285. S2CID 8032103.
Bennett, Michael A.; Martin, Greg; O'Bryant, Kevin; Rechnitzer, Andrew (2018). "Explicit bounds for primes in arithmetic progressions". Illinois J. Math. 62 (1–4): 427–532. arXiv:1802.00085. doi:10.1215/ijm/1552442669. S2CID 119647640.
Granville, Andrew; Martin, Greg (2006). "Prime number races" (PDF). American Mathematical Monthly. 113 (1): 1–33. doi:10.2307/27641834. JSTOR 27641834. MR 2202918.
Lemke Oliver, Robert J.; Soundararajan, Kannan (2016-08-02). "Unexpected biases in the distribution of consecutive primes". Proceedings of the National Academy of Sciences. 113 (31): E4446-54. arXiv:1603.03720. Bibcode:2016PNAS..113E4446L. doi:10.1073/pnas.1605366113. ISSN 0027-8424. PMC 4978288. PMID 27418603.
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. MR 0003018.
Arias de Reyna, Juan; Toulisse, Jérémy (2013). "The n-th prime asymptotically". Journal de théorie des nombres de Bordeaux. 25 (3): 521–555. arXiv:1203.5413. doi:10.5802/jtnb.847. MR 3179675. Zbl 1298.11093.
Dusart, Pierre (1999). "The $k$ th prime is greater than $k (log k + log log k - 1)$ for $k \geq 2$ ". Mathematics of Computation. 68 (225): 411–415. doi:10.1090/S0025-5718-99-01037-6. MR 1620223.
Platt, David (2015). "Computing $π (x)$ analytically". Mathematics of Computation. 84 (293): 1521–1535. arXiv:1203.5712. doi:10.1090/S0025-5718-2014-02884-6. MR 3315519. S2CID 119174627.
Chebolu, Sunil; Mináč, Ján (December 2011). "Counting Irreducible Polynomials over Finite Fields Using the Inclusion $π$ Exclusion Principle". Mathematics Magazine. 84 (5): 369–371. arXiv:1001.0409. doi:10.4169/math.mag.84.5.369. JSTOR 10.4169/math.mag.84.5.369. S2CID 115181186.

harvard.edu (Global: 18^th place; English: 17^th place)

ui.adsabs.harvard.edu

Erdős, Paul (July 1949). "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem" (PDF). Proceedings of the National Academy of Sciences of the U.S.A. 35 (7): 374–384. Bibcode:1949PNAS...35..374E. doi:10.1073/pnas.35.7.374. PMC 1063042. PMID 16588909 – via renyi.hu.
Lemke Oliver, Robert J.; Soundararajan, Kannan (2016-08-02). "Unexpected biases in the distribution of consecutive primes". Proceedings of the National Academy of Sciences. 113 (31): E4446-54. arXiv:1603.03720. Bibcode:2016PNAS..113E4446L. doi:10.1073/pnas.1605366113. ISSN 0027-8424. PMC 4978288. PMID 27418603.

illinois.edu (Global: 1,349^th place; English: 866^th place)

faculty.math.illinois.edu

Kevin Ford (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 (Global: 26^th place; English: 20^th place)

Costa Pereira, N. (August–September 1985). "A short proof of Chebyshev's theorem". American Mathematical Monthly. 92 (7): 494–495. doi:10.2307/2322510. JSTOR 2322510.
Nair, M. (February 1982). "On Chebyshev-type inequalities for primes". American Mathematical Monthly. 89 (2): 126–129. doi:10.2307/2320934. JSTOR 2320934.
Selberg, Atle (1949), "An Elementary Proof of the Prime-Number Theorem", Annals of Mathematics, 50 (2): 305–313, doi:10.2307/1969455, JSTOR 1969455, MR 0029410, S2CID 124153092
Newman, D.J. (1980). "Simple analytic proof of the prime number theorem". American Mathematical Monthly. 87 (9): 693–696. doi:10.2307/2321853. JSTOR 2321853. MR 0602825.
Zagier, Don (1997). "Newman's short proof of the prime number theorem". American Mathematical Monthly. 104 (8): 705–708. doi:10.2307/2975232. JSTOR 2975232. MR 1476753 – via maa.org.
Schoenfeld, Lowell (1976). "Sharper Bounds for the Chebyshev Functions $ϑ (x)$ and $ψ (x)$ . II". Mathematics of Computation. 30 (134): 337–360. doi:10.2307/2005976. JSTOR 2005976. MR 0457374.
Jørgensen, Bent; Martínez, José Raúl; Tsao, Min (1994). "Asymptotic behaviour of the variance function". Scandinavian Journal of Statistics. 21 (3): 223–243. JSTOR 4616314. MR 1292637.
Granville, Andrew; Martin, Greg (2006). "Prime number races" (PDF). American Mathematical Monthly. 113 (1): 1–33. doi:10.2307/27641834. JSTOR 27641834. MR 2202918.
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. MR 0003018.
Chebolu, Sunil; Mináč, Ján (December 2011). "Counting Irreducible Polynomials over Finite Fields Using the Inclusion $π$ Exclusion Principle". Mathematics Magazine. 84 (5): 369–371. arXiv:1001.0409. doi:10.4169/math.mag.84.5.369. JSTOR 10.4169/math.mag.84.5.369. S2CID 115181186.

maa.org (Global: 3,479^th place; English: 2,444^th place)

Zagier, Don (1997). "Newman's short proof of the prime number theorem". American Mathematical Monthly. 104 (8): 705–708. doi:10.2307/2975232. JSTOR 2975232. MR 1476753 – via maa.org.

nih.gov (Global: 4^th place; English: 4^th place)

ncbi.nlm.nih.gov

Erdős, Paul (July 1949). "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem" (PDF). Proceedings of the National Academy of Sciences of the U.S.A. 35 (7): 374–384. Bibcode:1949PNAS...35..374E. doi:10.1073/pnas.35.7.374. PMC 1063042. PMID 16588909 – via renyi.hu.
Lemke Oliver, Robert J.; Soundararajan, Kannan (2016-08-02). "Unexpected biases in the distribution of consecutive primes". Proceedings of the National Academy of Sciences. 113 (31): E4446-54. arXiv:1603.03720. Bibcode:2016PNAS..113E4446L. doi:10.1073/pnas.1605366113. ISSN 0027-8424. PMC 4978288. PMID 27418603.

pubmed.ncbi.nlm.nih.gov

Erdős, Paul (July 1949). "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem" (PDF). Proceedings of the National Academy of Sciences of the U.S.A. 35 (7): 374–384. Bibcode:1949PNAS...35..374E. doi:10.1073/pnas.35.7.374. PMC 1063042. PMID 16588909 – via renyi.hu.
Lemke Oliver, Robert J.; Soundararajan, Kannan (2016-08-02). "Unexpected biases in the distribution of consecutive primes". Proceedings of the National Academy of Sciences. 113 (31): E4446-54. arXiv:1603.03720. Bibcode:2016PNAS..113E4446L. doi:10.1073/pnas.1605366113. ISSN 0027-8424. PMC 4978288. PMID 27418603.

numdam.org (Global: 8,650^th place; English: 7,668^th place)

Hadamard, Jacques (1896), "Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques.", Bulletin de la Société Mathématique de France, 24, Société Mathématique de France: 199–220, archived from the original on 2024-09-10

psu.edu (Global: 207^th place; English: 136^th place)

citeseerx.ist.psu.edu

Harrison, John (2009). "Formalizing an analytic proof of the Prime Number Theorem". Journal of Automated Reasoning. 43 (3): 243–261. CiteSeerX 10.1.1.646.9725. doi:10.1007/s10817-009-9145-6. MR 2544285. S2CID 8032103.
Soprounov, Ivan (1998). "A short proof of the Prime Number Theorem for arithmetic progressions". Ohio: Cleveland State University. CiteSeerX 10.1.1.179.460.

renyi.hu (Global: low place; English: low place)

Erdős, Paul (July 1949). "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem" (PDF). Proceedings of the National Academy of Sciences of the U.S.A. 35 (7): 374–384. Bibcode:1949PNAS...35..374E. doi:10.1073/pnas.35.7.374. PMC 1063042. PMID 16588909 – via renyi.hu.

semanticscholar.org (Global: 11^th place; English: 8^th place)

api.semanticscholar.org

Selberg, Atle (1949), "An Elementary Proof of the Prime-Number Theorem", Annals of Mathematics, 50 (2): 305–313, doi:10.2307/1969455, JSTOR 1969455, MR 0029410, S2CID 124153092
Kevin Ford (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.
Timothy 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. S2CID 11013503.
von Koch, Helge (1901). "Sur la distribution des nombres premiers" [On the distribution of prime numbers]. Acta Mathematica (in French). 24 (1): 159–182. doi:10.1007/BF02403071. MR 1554926. S2CID 119914826.
Cornaros, Charalambos; Dimitracopoulos, Costas (1994). "The prime number theorem and fragments of PA" (PDF). Archive for Mathematical Logic. 33 (4): 265–281. doi:10.1007/BF01270626. MR 1294272. S2CID 29171246. Archived from the original (PDF) on 2011-07-21.
Avigad, Jeremy; Donnelly, Kevin; Gray, David; Raff, Paul (2008). "A formally verified proof of the prime number theorem". ACM Transactions on Computational Logic. 9 (1): 2. arXiv:cs/0509025. doi:10.1145/1297658.1297660. MR 2371488. S2CID 7720253.
Harrison, John (2009). "Formalizing an analytic proof of the Prime Number Theorem". Journal of Automated Reasoning. 43 (3): 243–261. CiteSeerX 10.1.1.646.9725. doi:10.1007/s10817-009-9145-6. MR 2544285. S2CID 8032103.
Bennett, Michael A.; Martin, Greg; O'Bryant, Kevin; Rechnitzer, Andrew (2018). "Explicit bounds for primes in arithmetic progressions". Illinois J. Math. 62 (1–4): 427–532. arXiv:1802.00085. doi:10.1215/ijm/1552442669. S2CID 119647640.
Platt, David (2015). "Computing $π (x)$ analytically". Mathematics of Computation. 84 (293): 1521–1535. arXiv:1203.5712. doi:10.1090/S0025-5718-2014-02884-6. MR 3315519. S2CID 119174627.
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Lemke Oliver, Robert J.; Soundararajan, Kannan (2016-08-02). "Unexpected biases in the distribution of consecutive primes". Proceedings of the National Academy of Sciences. 113 (31): E4446-54. arXiv:1603.03720. Bibcode:2016PNAS..113E4446L. doi:10.1073/pnas.1605366113. ISSN 0027-8424. PMC 4978288. PMID 27418603.

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