Archive for History of Exact Sciences

, Volume 68, Issue 3, pp 265–326 | Cite as

The concept of “character” in Dirichlet’s theorem on primes in an arithmetic progression

  • Jeremy Avigad
  • Rebecca Morris


In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.


  1. Avigad, Jeremy. Methodology and metaphysics in the development of Dedekind’s theory of ideals. In (Ferreirós and Gray, 2006), 159–186.Google Scholar
  2. Avigad, Jeremy. 2006. Mathematical method and proof. Synthese 153: 105–159.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Avigad, Jeremy. 2008. Understanding proofs. In (Mancosu, 2008), 317–353.Google Scholar
  4. Avigad, Jeremy. 2010. Understanding, formal verification, and the philosophy of mathematics. Journal of the Indian Council of Philosophical Research 27:161–197.Google Scholar
  5. Avigad, Jeremy, and Rebecca Morris. n.d. Character and object. In preparation.Google Scholar
  6. Beaney, Michael (ed.). 1997. The Frege Reader. Malden, MA: Blackwell Publishing.Google Scholar
  7. Boole, George. 1859. A treatise on differential equations. Cambridge: Macmillan and Co.Google Scholar
  8. Bottazzini, Umberto, and Jeremy Gray. 2013. Hidden Harmony-Geometric Fantasies: The rise of complex function theory. Berlin: Springer.CrossRefGoogle Scholar
  9. Cantor, Georg. 1895. Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische Annalen 46: 481–512.CrossRefzbMATHGoogle Scholar
  10. Cayley, Arthur. 1854. On the theory of groups, as depending on the symbolic equation \(\theta ^n = 1\). Philosophical Magazine 7:40–47. Reprinted in his Collected Mathematical Papers, volume 2, 123–130. Cambridge: Cambridge University Press. 1889.Google Scholar
  11. Chorlay, Renaud. n.d. Questions of generality as probes into 19th century analysis. To appear in K. Chemla, R. Chorlay, D. Rabouin, editors, Handbook on Generality in Mathematics and the Sciences.Google Scholar
  12. de la Vallée Poussin, Charles Jean. Démonstration simplifiée du théorèm de Dirichlet sur la progression arithmétique. Mémoires couronnés et autres mémoires publiés par L’Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique 53:1895–1896.Google Scholar
  13. de la Vallée Poussin, Charles Jean. 1897. Recherches analytiques sur la théorie des nombres premiers. Brussels: Hayez.Google Scholar
  14. Dedekind, Richard. 1854. Über die Einführung neuer Funktionen in der Mathematik. Delivered as a Hablitationsvorlesung in Göttingen on June 30, 1854. Reprinted in (Dedekind, 1932), volume 3, chapter LX, 428–438, and translated by William Ewald as “On the introduction of new functions in mathematics” in (Ewald, 1996), volume 2, 754–762.Google Scholar
  15. Dedekind, Richard. 1888. Was sind und was sollen die Zahlen? Braunschweig: F. Vieweg & Sohn. A later edition reprinted in (Dedekind, 1932), Volume 3, Chapter LI, 335–391. The second edition, with a new preface, was published in 1893, and is translated by Wooster Beman as “The nature and meaning of numbers” in Essays on the Theory of Numbers, Chicago: Open Court, 1901; reprinted by New York: Dover, 1963. The Beman translation is reprinted, with corrections by William Ewald, in (Ewald, 1996), volume 2, 787–833.Google Scholar
  16. Dedekind, Richard. 1894. Über die Theorie der ganzen algebraischen Zahlen. Supplement XI to the fourth edition of (Dirichelt, 1863), 434–657. Reprinted in (Dedekind, 1932), volume 3, chapter 46, 1–222.Google Scholar
  17. Dedekind, Richard. 1932. Gesammelte mathematische Werke, volumes 1–3, edited by Robert Fricke, Emmy Noether and Öystein Ore, Braunschweig: F. Vieweg & Sohn. Reprinted by Chelsea Publishing Co., New York, 1968.Google Scholar
  18. Dirichlet, Johann Peter Gustav Lejeune. 1829. Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données. Journal für die reine und angewandte Mathematik, 4:157–169. Reprinted in (Dirichlet, 1889), 117–132.Google Scholar
  19. Dirichlet, Johann Peter Gustave Lejeune. 1835. Ueber eine neue anwendung bestimmter integrale auf die summation endlicher oder unendlicher reihen. Abhandlungen der königlich Preussischen Akademie der Wissenschaften, 391–407. Reprinted in (Dirichlet, 1889), 237–256.Google Scholar
  20. Dirichlet, Johann Peter Gustav Lejeune. 1837. Beweis eines Satzes über die arithmetische Progression. Bericht über die Verhandlungen der königlich Presussischen Akademie der Wissenschaften Berlin. Reprinted in (Dirichlet, 1889), pages 307–312.Google Scholar
  21. Dirichlet, Johann Peter Gustav Lejeune. 1837. Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der königlich Preussischen Akademie der Wissenschaften 45–81. Reprinted in (Dirichlet, 1889), 313–342. Translated by Ralf Stefan as “There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime”, arxiv:0808.1408.Google Scholar
  22. Dirichlet, Johann Peter Gustav Lejeune. 1840. Über eine Eigenschaft der quadratischen Formen. Journal für die reine und angewandte Mathematik, 21:98–100. Reprinted in (Dirichlet, 1889), 597–502.Google Scholar
  23. Dirichlet, Johann Peter Gustav Lejeune. 1841. Untersuchungen über die Theorie der complexen Zahlen. Journal für die reine und angewandte Mathematik 22:190–194. Reprinted in (Dirichlet, 1889), 503–508.Google Scholar
  24. Dirichlet, Johann Peter Gustave Lejeune. 1863. Vorlsesungen über Zahlentheorie. Vieweg, Braunschweig. Edited by Richard Dedekind. Subsequent editions in 1871, 1879, 1894, with “supplements” by Richard Dedekind. Translated by John Stillwell, with introductory notes, as Lectures on Number Theory, American Mathematical Society, Providence, RI, 1999.Google Scholar
  25. Dirichlet, Johann Peter Gustav Lejune. 1889. Werke, edited by Leopold Kronecker. Berlin: Georg Reimer.Google Scholar
  26. Edwards, Harold M. 1980. The genesis of ideal theory. Archive for history of exact sciences 23: 321–378.CrossRefzbMATHMathSciNetGoogle Scholar
  27. Edwards, Harold M. 1989. Kronecker’s views on the foundations of mathematics. In The history of modern mathematics, ed. D.E. Rowe, and J. McCleary, 67–77. San Diego: Academic Press.Google Scholar
  28. Edwards, Harold M. 2007. Kronecker’s fundamental theorem of general arithmetic. In Episodes in the history of modern algebra (1800–1950), ed. Jeremy Gray, and Karen Parshall, 107–116. Providence, RI: American Mathematical Society.Google Scholar
  29. Edwards, Harold M. 2009. Kronecker’s algorithmic mathematics. Mathematical Intelligencer 31: 11–14.CrossRefzbMATHMathSciNetGoogle Scholar
  30. Eistenstein, Gotthold. 1850. Eine neue Gattung zahlentheoretischer Funktionen, weche von zwei Elementen abhängen und durch gewisse lineare Funktional-Gleichungen definirt werden. Bericht über die zur Bekanntmachung geeigneten Verhandlungen der königlich Preussischen Akademie der Wissenshaften zu Berlin, 36–42. Reprinted in Eisenstein’s Mathematische Werke, volume 2, pages 705–711, Chelsea Publishing Company, New York, 1989.Google Scholar
  31. Euler, Leonhard. 1748. Introductio in Analysin Infinitorum, tomus primus. Lausannae. Publications E101 and E102 in the Euler Archive.Google Scholar
  32. Euler, Leonhard. 1784. Speculationes circa quasdam insignes proprietates numerorum. Acta Academiae Scientarum Imperialis Petropolitinae 4. Publication E564 in the Euler Archive. Translated by Jordan Bell as “Speculations on some characteristic properties of numbers”, 2007, arXiv:0705.3929.Google Scholar
  33. Everest, Graham, and Thomas Ward. 2005. An introduction to number theory. London: Springer.zbMATHGoogle Scholar
  34. Ewald, William (ed.). 1996. From Kant to Hilbert: A source book in the foundations of mathematics, volumes 1 and 2. Oxford: Oxford University Press.Google Scholar
  35. Ferreirós, José. 1999. Labyrinth of thought: A history of set theory and its role in modern mathematics. Basel: Birkhäuser.CrossRefzbMATHGoogle Scholar
  36. Ferreirós, José, and Gray Jeremy (eds.). 2006. The architecture of modern mathematics. Oxford: Oxford University Press.zbMATHGoogle Scholar
  37. Frege, Gottlob. 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Louis Nebert. Translated by Michael Beaney in (Beaney, 1997), 84–129.Google Scholar
  38. Frege, Gottlob. 1891. Function und Begriff. Jena: Hermann Pohle. Reprinted in (Frege, 2002), and translated by Peter Geach as “Function and concept” in (Beaney, 1997). Page number references are to the original publication.Google Scholar
  39. Frege, Gottlob. 1893. Grundgesetze der Arithmetik. Jena: H. Pohle, volume 1, volume 2, 1903. Excerpts translated by Michael Beaney in (Beaney, 1997), 194–223 and 258–289.Google Scholar
  40. Frege, Gottlob. 2002. Funktion-Begriff-Bedeutung, edited by Mark Textor. Göttingen: Vandenhoeck and Ruprecht.Google Scholar
  41. Gauss, Carl Friedrich. 1801. Disquisitiones Arithmeticae. Leipzig: G. Fleischer. Reprinted in Gauss’ Werke, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1863. Translated with a preface by Arthur A. Clarke, New Haven: Yale University Press, 1966, and republished by New York: Springer, 1986.Google Scholar
  42. Hadamard, Jacques. 1896. Sur la distribution des zéros de la fonction \(\zeta (s)\) et ses conséquences arithmétiques. Bulletin de la Société Mathématique de France 24: 199–220.Google Scholar
  43. Harel, Guershon, Evan Fuller, and Jeffrey M. Rabin. 2008. Attention to meaning by algebra teachers. Journal of Mathematical Behavior 27: 116–127.CrossRefGoogle Scholar
  44. Harel, Guershon, and James Kaput. 1991. The role of conceptual entities in building advanced mathematical concepts and their symbols. In Advanced mathematical thinking, ed. D. Tall, 82–94. Dordrecht: Kluwer.Google Scholar
  45. Hausdorff, Felix. 1914. Grundzüge der Mengenlehre. Leipzig: Veit and Company. Reprinted by Chelsea Publishing Company, New York, 1949.Google Scholar
  46. Hawkins, Thomas. 1971. The origins of the theory of group characters. Archive for History of Exact Sciences 7: 142–170.CrossRefMathSciNetGoogle Scholar
  47. Klein, Felix. 1893. A comparative review of recent researches in geometry. Bulletin of the American Mathematical Society 2: 215–249.CrossRefzbMATHMathSciNetGoogle Scholar
  48. Kleiner, Israel. 1989. Evolution of the function concept: A brief survey. The College Mathematical Journal 20: 282–300.CrossRefGoogle Scholar
  49. Kronecker, Leopold. 1870. Auseinandersetzung einiger eigenschaften der klassenzahl idealer complexer zahlen. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 881–882. Reproduced in (Kronecker, 1895–1930), volume I, pages 271–282.Google Scholar
  50. Kronecker, Leopold. 1882. Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Riemer, Berlin. Also published in Journal für reine und angewandte Mathematik, volume 92, 1–122, and (Kronecker, 1895–1930), volume II, 237–387.Google Scholar
  51. Kronecker, Leopold. 1887. Ein Fundamentalsatz der allgemeinen Arithmetik. Journal für die reine und angewandte Mathematik 100:490–510. Reprinted in (Kronecker, 1895–1930), volume IIIa, pages 209–240.Google Scholar
  52. Kronecker, Leopold. 1901. Vorlesungen über Zahlentheorie, edited by Kurt Hensel, B. G. Teubner, Leipzig. Republished by Springer, Berlin, 1978.Google Scholar
  53. Kronecker, Leopold. 1895–1930. Werke, edited by Kurt Hensel, volumes 1–5, B. G. Teubner, Leipzig. Reprinted by Chelsea Publishing Co., New York.Google Scholar
  54. Landau, Edmund. 1909. Handbuch der Lehre von der Verteilung der Primzahlen, volume 1. Leipzig: B. G. Teubner.Google Scholar
  55. Landau, Edmund. 1927. Vorlesungen über Zahlentheorie. S. Hirzel, Leipzig. Translated by Jacob E. Goodman, as Elementary number theory, Chelsea Publishing Company, New York.Google Scholar
  56. Legendre, Adrien-Marie. 1788. Recherches d’analyse indéterminée. Histoire de l’Academie Royale des Sciences de Paris, 465–559.Google Scholar
  57. Luzin, Nikolai. 1998. Functions. The American Mathematical Monthly 105:59–67 (Part I) and 105:263–270 (Part II).Google Scholar
  58. Mackey, George W. 1980. Harmonic analysis as the exploitation of symmetry: A historical survey. Bulletin of the American Mathematical Society 3: 543–698.CrossRefzbMATHMathSciNetGoogle Scholar
  59. Mancosu, Paolo (ed.). 2008. The philosophy of mathematical practice. Oxford: Oxford University Press.zbMATHGoogle Scholar
  60. Monna, A.F. 1972. The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue. Archive for History of Exact Sciences 9: 57–84.CrossRefzbMATHMathSciNetGoogle Scholar
  61. Moore, Gregory H. 1982. Zermelo’s axiom of choice: Its origins, development, and influence. New York: Springer.CrossRefzbMATHGoogle Scholar
  62. Morris, Rebecca. 2011. Character and object. Master’s thesis. Carnegie Mellon University, PittsburghGoogle Scholar
  63. Pengelley, David. 2005. Arthur Cayley and the first paper on group theory. In From Calculus to Computers: Using the Last 200 Years of Mathematics History in the Classroom, ed. Amy Shell-Gellasch, and Dick Jardine, 3–8. Washington, D.C.: Mathematics Association of America.Google Scholar
  64. Stein, Howard. 1988. Logos, logic, and logistiké. In History and Philosophy of Modern Mathematics, ed. William Aspray, and Philip Kitcher, 238–259. Minneapolis, MN: University of Minnesota Press.Google Scholar
  65. Tappenden, Jamie. 2006. The Riemannian background to Frege’s philosophy. In (Ferreirós and Gray, 2006), 97–132.Google Scholar
  66. Troelstra, A.S., Dirk van Dalen. 1988. Constructivism in mathematics: An introduction, volumes 1 and 2. Amsterdam: North-Holland.Google Scholar
  67. Volterra, Vito. 1887. Sopra le funzioni che dipendono de altre funzioni. Rend. R. Academia dei Lincei 2:97–105, 141–146 and 153–158.Google Scholar
  68. Volterra, Vito. 1930. Theory of Functionals and of Integral and Integro-differential Equations. Blackie and Son, Ltd., London and Glasgow. Translated by M. Long and edited by Luigi Fantappiè. Reprinted with a preface by G. C. Evans, a biography of Vito Volterra and a bibliography of his published works by E. Whittaker, New York: Dover, 1959.Google Scholar
  69. Weber, Heinrich. 1882. Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist. Mathematische Annalen 20: 301–329.CrossRefzbMATHMathSciNetGoogle Scholar
  70. Wilson, Mark. 2006. Wandering significance: An essay on conceptual behavior. Oxford: Oxford University Press.CrossRefGoogle Scholar
  71. Wussing, Hans. 1984. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, translated from the German by Abe Shenitzer and Hardy Grant. Cambridge, MA: MIT Press.Google Scholar
  72. Youschkevitch, A.P. 1976. The concept of function up to the middle of the 19th century. Archive for History of Exact Sciences 16: 37–85.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Philosophy and Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of PhilosophyCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations