Abstract
In this section, we briefly introduce Hurwitz’s Hurwitz generalization of Bernoulli numbers, known as the Hurwitz numbers.
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Notes
- 1.
Niels Henrik Abel (born on August 5, 1802 in Frindoe, Norway—died on April 6, 1829 in Froland, Norway).
- 2.
- 3.
Hermann Cäsar Hannibal Schubert (born on May 22, 1848 in Potsdam, Germany—died on July 20, 1911 in Hamburg, Germany).
- 4.
Felix Christian Klein (born on April 25, 1849 in Düsseldorf, Prussia (now Germany)—died on June 22, 1925 in Göttingen, Germany).
- 5.
Carl Louis Ferdinand von Lindemann (born on April 12, 1852 in Hannover, Hanover (now Germany)—died on March 6, 1939 in Munich, Germany) who proved in 1882 that π is a transcendental number .
- 6.
David Hilbert (born on January 23, 1862 in Königsberg, Prussia (now Kaliningrad, Russia)—died on February 14, 1943 in Göttingen, Germany).
- 7.
Hermann Minkowski (born on June 22, 1864 in Alexotas, Russian Empire (now Kaunas, Lithuania)—died on January 12, 1909 in Göttingen, Germany).
- 8.
Ferdinand Georg Frobenius (born on October 26, 1849 in Berlin-Charlottenburg, Prussia (now Germany)—died on August 3, 1917 in Berlin, Germany).
- 9.
George Pólya (born on December 13, 1887 in Budapest, Hungary—died on September 7, 1985 in Palo Alto, USA).
References
Coates, J., Wiles, A.: Kummer’s criterion for Hurwitz numbers, in Algebraic number theory (Kyoto Internat. Sympos., RIMS, Kyoto, 1976), pp. 9–23. Japan Soc. Promotion Sci., Tokyo (1977)
Freudenthal, H.: Hurwitz, Adolf, in Dictionary of Scientific Biography, vols. 5 & 6. Charles Scribner’s Sons, New York (1981)
Gauss, C.F.: Disquisitiones Arithmeticae (1801)
Hilbert, D.: Adolf Hurwitz, Gedächtnisrede, Nachrichten von der k. Gesellshaft der Wissenschaften zu Göttingen (1920), pp. 75–83 (Hurwitz’s Mathematische Werke, I xiii–xx)
Hurwitz, A.: Über die Entwicklungskoeffizienten der lemniskatischen Funktionen. Math. Ann. 51, 196–226 (1899). (Mathematische Werke II, 342–373)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer (1990)
Katz, N.: The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216, 1–4 (1975)
Pólya, G.: Some mathematicians I have known. Am. Math. Monthly 76, 746–753 (1969)
Rademacher, H.: Lectures on Elementary Number Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 169. Springer (1773)
Reid, C.: Hilbert. Springer, Berlin-Heidelberg-New York (1970)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927)
Zagier, D.: A one-sentence proof that every prime \(p \equiv 1\pmod 4\) is a sum of two squares. Am. Math. Monthly 97–2, 144 (1990)
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Ibukiyama, T., Kaneko, M. (2014). Hurwitz Numbers. In: Bernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54919-2_12
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