Riemann hypothesis (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Riemann hypothesis" in English language version.

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Ingham (1932), Theorem 30, p. 83; Montgomery & Vaughan (2007), p. 430. Ingham, A.E. (1932), The Distribution of Prime Numbers, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 30, Cambridge University Press. Reprinted 1990, ISBN 978-0-521-39789-6, MR 1074573 Montgomery, Hugh L.; Vaughan, Robert C. (2007), Multiplicative Number Theory I. Classical Theory, Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press.ISBN 978-0-521-84903-6
Ingham (1932), p. 82. Ingham, A.E. (1932), The Distribution of Prime Numbers, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 30, Cambridge University Press. Reprinted 1990, ISBN 978-0-521-39789-6, MR 1074573
Robin (1984). Robin, G. (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées, Neuvième Série, 63 (2): 187–213, MR 0774171
Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008, S2CID 15884740
Titchmarsh (1986). Titchmarsh, Edward Charles (1986), The theory of the Riemann zeta-function (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853369-6, MR 0882550
Nicely (1999). Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation, 68 (227): 1311–1315, Bibcode:1999MaCom..68.1311N, doi:10.1090/S0025-5718-99-01065-0, MR 1627813.
Rodgers & Tao (2020). Rodgers, Brad; Tao, Terence (2020), "The de Bruijn–Newman constant is non-negative", Forum of Mathematics, 8: e6, 62, arXiv:1801.05914, doi:10.1017/fmp.2020.6, MR 4089393; see also announcement on Tao's blog, January 19, 2018
Radziejewski (2007). Radziejewski, Maciej (2007), "Independence of Hecke zeta functions of finite order over normal fields", Transactions of the American Mathematical Society, 359 (5): 2383–2394, doi:10.1090/S0002-9947-06-04078-5, MR 2276625, There are infinitely many nonisomorphic algebraic number fields whose Dedekind zeta functions have infinitely many nontrivial multiple zeros.
Wiles (2000). Wiles, Andrew (2000), "Twenty years of number theory", Mathematics: frontiers and perspectives, Providence, R.I.: American Mathematical Society, pp. 329–342, ISBN 978-0-8218-2697-3, MR 1754786
Leichtnam (2005). Leichtnam, Eric (2005), "An invitation to Deninger's work on arithmetic zeta functions", Geometry, spectral theory, groups, and dynamics, Contemp. Math., vol. 387, Providence, RI: Amer. Math. Soc., pp. 201–236, doi:10.1090/conm/387/07243, ISBN 978-0-8218-3710-8, MR 2180209.
Knauf (1999). Knauf, Andreas (1999), "Number theory, dynamical systems and statistical mechanics", Reviews in Mathematical Physics, 11 (8): 1027–1060, Bibcode:1999RvMaP..11.1027K, doi:10.1142/S0129055X99000325, MR 1714352
Edwards (1974). Edwards, H. M. (1974), Riemann's Zeta Function, New York: Dover Publications, ISBN 978-0-486-41740-0, MR 0466039
Lehmer (1956). Lehmer, D. H. (1956), "Extended computation of the Riemann zeta-function", Mathematika, 3 (2): 102–108, doi:10.1112/S0025579300001753, MR 0086083

arxiv.org

Soundararajan, Kannan (2009), "Partial sums of the Möbius function", J. Reine Angew. Math., 2009 (631): 141–152, arXiv:0705.0723, doi:10.1515/CRELLE.2009.044, S2CID 16501321
Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008, S2CID 15884740
Baez-Duarte, Luis (2005). "A general strong Nyman-Beurling criterion for the Riemann hypothesis". Publications de l'Institut Mathématique. Nouvelle Série. 78 (92): 117–125. arXiv:math/0505453. doi:10.2298/PIM0578117B. S2CID 17406178.
Rodgers & Tao (2020). Rodgers, Brad; Tao, Terence (2020), "The de Bruijn–Newman constant is non-negative", Forum of Mathematics, 8: e6, 62, arXiv:1801.05914, doi:10.1017/fmp.2020.6, MR 4089393; see also announcement on Tao's blog, January 19, 2018
Platt & Trudgian (2021). Platt, Dave; Trudgian, Tim (January 2021), "The Riemann hypothesis is true up to $3\cdot 10^{12}$ ", Bulletin of the London Mathematical Society, 53 (3), Wiley: 792–797, arXiv:2004.09765, doi:10.1112/blms.12460, S2CID 234355998
Dunn, Alexander; Radziwiłł, Maksym (2021). "Bias in cubic Gauss sums: Patterson's conjecture". arXiv:2109.07463 [math.NT].
Mossinghoff, Michael J.; Trudgian, Timothy S.; Yang, Andrew (2022-12-13). "Explicit zero-free regions for the Riemann zeta-function". arXiv:2212.06867 [math.NT].
Pratt, Kyle; Robles, Nicolas; Zaharescu, Alexandru; Zeindler, Dirk (2020). "More than five-twelfths of the zeros of ζ are on the critical line". Res Math Sci. 7. arXiv:1802.10521. doi:10.1007/s40687-019-0199-8. S2CID 202542332.
Johnston, David R. (29 July 2022). "Improving bounds on prime counting functions by partial verification of the Riemann hypothesis". The Ramanujan Journal. 59 (4): 1307–1321. arXiv:2109.02249. doi:10.1007/s11139-022-00616-x. S2CID 237420836.

caltech.edu

"Caltech Mathematicians Solve 19th Century Number Riddle". California Institute of Technology. October 31, 2022.

claymath.org

Bombieri (2000). Bombieri, Enrico (2000), The Riemann Hypothesis – official problem description (PDF), Clay Mathematics Institute, archived from the original (PDF) on 2015-12-22, retrieved 2008-10-25 Reprinted in (Borwein et al. 2008).
Sarnak (2005). Sarnak, Peter (2005), Problems of the Millennium: The Riemann Hypothesis (2004) (PDF), Clay Mathematics Institute, retrieved 2015-07-28 Reprinted in (Borwein et al. 2008).

doi.org

Landau, Edmund (1924), "Über die Möbiussche Funktion", Rend. Circ. Mat. Palermo, 48 (2): 277–280, doi:10.1007/BF03014702, S2CID 123636883
Titchmarsh, Edward Charles (1927), "A consequence of the Riemann hypothesis", J. London Math. Soc., 2 (4): 247–254, doi:10.1112/jlms/s1-2.4.247
Maier, Helmut; Montgomery, Hugh (2009), "The sum of the Möbius function", Bull. London Math. Soc., 41 (2): 213–226, doi:10.1112/blms/bdn119, hdl:2027.42/135214, S2CID 121272525
Soundararajan, Kannan (2009), "Partial sums of the Möbius function", J. Reine Angew. Math., 2009 (631): 141–152, arXiv:0705.0723, doi:10.1515/CRELLE.2009.044, S2CID 16501321
Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008, S2CID 15884740
Nicely (1999). Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation, 68 (227): 1311–1315, Bibcode:1999MaCom..68.1311N, doi:10.1090/S0025-5718-99-01065-0, MR 1627813.
Baez-Duarte, Luis (2005). "A general strong Nyman-Beurling criterion for the Riemann hypothesis". Publications de l'Institut Mathématique. Nouvelle Série. 78 (92): 117–125. arXiv:math/0505453. doi:10.2298/PIM0578117B. S2CID 17406178.
Rodgers & Tao (2020). Rodgers, Brad; Tao, Terence (2020), "The de Bruijn–Newman constant is non-negative", Forum of Mathematics, 8: e6, 62, arXiv:1801.05914, doi:10.1017/fmp.2020.6, MR 4089393; see also announcement on Tao's blog, January 19, 2018
Platt & Trudgian (2021). Platt, Dave; Trudgian, Tim (January 2021), "The Riemann hypothesis is true up to $3\cdot 10^{12}$ ", Bulletin of the London Mathematical Society, 53 (3), Wiley: 792–797, arXiv:2004.09765, doi:10.1112/blms.12460, S2CID 234355998
Goldfeld, Dorian (1985). "Gauss' class number problem for imaginary quadratic fields". Bulletin of the American Mathematical Society. 13 (1): 23–37. doi:10.1090/S0273-0979-1985-15352-2. ISSN 0273-0979.
Siegel, Carl (1935). "Über die Classenzahl quadratischer Zahlkörper". Acta Arithmetica. 1 (1): 83–86. doi:10.4064/aa-1-1-83-86. ISSN 0065-1036. Retrieved 8 April 2024.
Radziejewski (2007). Radziejewski, Maciej (2007), "Independence of Hecke zeta functions of finite order over normal fields", Transactions of the American Mathematical Society, 359 (5): 2383–2394, doi:10.1090/S0002-9947-06-04078-5, MR 2276625, There are infinitely many nonisomorphic algebraic number fields whose Dedekind zeta functions have infinitely many nontrivial multiple zeros.
Leichtnam (2005). Leichtnam, Eric (2005), "An invitation to Deninger's work on arithmetic zeta functions", Geometry, spectral theory, groups, and dynamics, Contemp. Math., vol. 387, Providence, RI: Amer. Math. Soc., pp. 201–236, doi:10.1090/conm/387/07243, ISBN 978-0-8218-3710-8, MR 2180209.
Knauf (1999). Knauf, Andreas (1999), "Number theory, dynamical systems and statistical mechanics", Reviews in Mathematical Physics, 11 (8): 1027–1060, Bibcode:1999RvMaP..11.1027K, doi:10.1142/S0129055X99000325, MR 1714352
Pratt, Kyle; Robles, Nicolas; Zaharescu, Alexandru; Zeindler, Dirk (2020). "More than five-twelfths of the zeros of ζ are on the critical line". Res Math Sci. 7. arXiv:1802.10521. doi:10.1007/s40687-019-0199-8. S2CID 202542332.
Johnston, David R. (29 July 2022). "Improving bounds on prime counting functions by partial verification of the Riemann hypothesis". The Ramanujan Journal. 59 (4): 1307–1321. arXiv:2109.02249. doi:10.1007/s11139-022-00616-x. S2CID 237420836.
Lehmer (1956). Lehmer, D. H. (1956), "Extended computation of the Riemann zeta-function", Mathematika, 3 (2): 102–108, doi:10.1112/S0025579300001753, MR 0086083

eudml.org

Siegel, Carl (1935). "Über die Classenzahl quadratischer Zahlkörper". Acta Arithmetica. 1 (1): 83–86. doi:10.4064/aa-1-1-83-86. ISSN 0065-1036. Retrieved 8 April 2024.

handle.net

hdl.handle.net

Maier, Helmut; Montgomery, Hugh (2009), "The sum of the Möbius function", Bull. London Math. Soc., 41 (2): 213–226, doi:10.1112/blms/bdn119, hdl:2027.42/135214, S2CID 121272525

harvard.edu

ui.adsabs.harvard.edu

Nicely (1999). Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation, 68 (227): 1311–1315, Bibcode:1999MaCom..68.1311N, doi:10.1090/S0025-5718-99-01065-0, MR 1627813.
Knauf (1999). Knauf, Andreas (1999), "Number theory, dynamical systems and statistical mechanics", Reviews in Mathematical Physics, 11 (8): 1027–1060, Bibcode:1999RvMaP..11.1027K, doi:10.1142/S0129055X99000325, MR 1714352

jstor.org

Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008, S2CID 15884740

maa.org

eulerarchive.maa.org

Euler, Leonhard (1744). Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae 9, pp. 160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case $s=1$ , and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that $\zeta (1)=\log \infty$ , and he makes use of this latter result in his Theorem 19, to show that the sum of the inverses of the prime numbers is $\log \log \infty$ .

semanticscholar.org

api.semanticscholar.org

Landau, Edmund (1924), "Über die Möbiussche Funktion", Rend. Circ. Mat. Palermo, 48 (2): 277–280, doi:10.1007/BF03014702, S2CID 123636883
Maier, Helmut; Montgomery, Hugh (2009), "The sum of the Möbius function", Bull. London Math. Soc., 41 (2): 213–226, doi:10.1112/blms/bdn119, hdl:2027.42/135214, S2CID 121272525
Soundararajan, Kannan (2009), "Partial sums of the Möbius function", J. Reine Angew. Math., 2009 (631): 141–152, arXiv:0705.0723, doi:10.1515/CRELLE.2009.044, S2CID 16501321
Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008, S2CID 15884740
Baez-Duarte, Luis (2005). "A general strong Nyman-Beurling criterion for the Riemann hypothesis". Publications de l'Institut Mathématique. Nouvelle Série. 78 (92): 117–125. arXiv:math/0505453. doi:10.2298/PIM0578117B. S2CID 17406178.
Platt & Trudgian (2021). Platt, Dave; Trudgian, Tim (January 2021), "The Riemann hypothesis is true up to $3\cdot 10^{12}$ ", Bulletin of the London Mathematical Society, 53 (3), Wiley: 792–797, arXiv:2004.09765, doi:10.1112/blms.12460, S2CID 234355998
Pratt, Kyle; Robles, Nicolas; Zaharescu, Alexandru; Zeindler, Dirk (2020). "More than five-twelfths of the zeros of ζ are on the critical line". Res Math Sci. 7. arXiv:1802.10521. doi:10.1007/s40687-019-0199-8. S2CID 202542332.
Johnston, David R. (29 July 2022). "Improving bounds on prime counting functions by partial verification of the Riemann hypothesis". The Ramanujan Journal. 59 (4): 1307–1321. arXiv:2109.02249. doi:10.1007/s11139-022-00616-x. S2CID 237420836.

springer.com

link.springer.com

Johnston, David R. (29 July 2022). "Improving bounds on prime counting functions by partial verification of the Riemann hypothesis". The Ramanujan Journal. 59 (4): 1307–1321. arXiv:2109.02249. doi:10.1007/s11139-022-00616-x. S2CID 237420836.

terrytao.wordpress.com

Rodgers & Tao (2020). Rodgers, Brad; Tao, Terence (2020), "The de Bruijn–Newman constant is non-negative", Forum of Mathematics, 8: e6, 62, arXiv:1801.05914, doi:10.1017/fmp.2020.6, MR 4089393; see also announcement on Tao's blog, January 19, 2018

trnicely.net

Nicely (1999). Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation, 68 (227): 1311–1315, Bibcode:1999MaCom..68.1311N, doi:10.1090/S0025-5718-99-01065-0, MR 1627813.

umn.edu

dtc.umn.edu

Odlyzko (2002). Odlyzko, Andrew (2002), Zeros of the Riemann zeta function: Conjectures and computations (PDF) Slides of a talk

web.archive.org

Bombieri (2000). Bombieri, Enrico (2000), The Riemann Hypothesis – official problem description (PDF), Clay Mathematics Institute, archived from the original (PDF) on 2015-12-22, retrieved 2008-10-25 Reprinted in (Borwein et al. 2008).

wolfram.com

mathworld.wolfram.com

Weisstein, Eric W., "Riemann Zeta Function Zeros", MathWorld: "ZetaGrid is a distributed computing project attempting to calculate as many zeros as possible. It had reached 1029.9 billion zeros as of Feb. 18, 2005."

wolframalpha.com

Values for ζ can be found by calculating, e.g., ζ(1/2 − 30i).("Wolframalpha computational intelligence". wolframalpha.com. Wolfram. Retrieved 2 October 2022.

worldcat.org

search.worldcat.org

Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008, S2CID 15884740
Goldfeld, Dorian (1985). "Gauss' class number problem for imaginary quadratic fields". Bulletin of the American Mathematical Society. 13 (1): 23–37. doi:10.1090/S0273-0979-1985-15352-2. ISSN 0273-0979.
Siegel, Carl (1935). "Über die Classenzahl quadratischer Zahlkörper". Acta Arithmetica. 1 (1): 83–86. doi:10.4064/aa-1-1-83-86. ISSN 0065-1036. Retrieved 8 April 2024.