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Rademacher Sums and Rademacher Series

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Conformal Field Theory, Automorphic Forms and Related Topics

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 8))

Abstract

We exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics.

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Notes

  1. 1.

    Norton, in unpublished work (cf. [16]), has found 616 groups Γ such that \(\varGamma _{\infty } =\langle T,-I\rangle\), the congruence group Γ 0(N) is contained in Γ for some N, and the coefficients of the corresponding Rademacher sum R Γ, 1, 0 [−1] are rational, and Cummins has shown [16] that 6,486 genus zero groups are obtained by dropping the condition of rationality. On the other hand, there are 194 conjugacy classes in the monster, but the two classes of order 27 are related by inversion and thus determine the same McKay–Thompson series. There are no other coincidences amongst the T g but there are some linear relations, and curiously, the space of functions spanned linearly by the T g for \(g \in \mathbb{M}\) is 163 dimensional.

References

  1. Andrews, G.E.: Ramunujan’s “lost” notebook. III. The Rogers-Ramanujan continued fraction. Adv. Math. 41(2), 186–208 (1981)

    MATH  Google Scholar 

  2. Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics, vol. 41, 2nd edn. Springer, New York (1990)

    Google Scholar 

  3. Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109(2), 405–444 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bringmann, K., Ono, K.: The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165(2), 243–266 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bringmann, K., Ono, K.: Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series. Math. Ann. 337(3), 591–612 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bringmann, K., Ono, K.: Dyson’s ranks and Maass forms. Ann. Math. (2) 171(1), 419–449 (2010)

    Google Scholar 

  7. Bringmann, K., Ono, K.: Coefficients of harmonic maass forms. In: Alladi, K., Garvan, F. (eds.) Partitions, q-Series, and Modular Forms. Developments in Mathematics, vol. 23, pp. 23–38. Springer, New York (2012)

    Chapter  Google Scholar 

  8. Carnahan, S.: Monstrous Lie algebras and generalized moonshine. Ph.D. thesis, University of California, Berkeley (2007)

    Google Scholar 

  9. Cho, B., Choie, Y.: Zagier duality for harmonic weak Maass forms of integral weight. Proc. Am. Math. Soc. 139(3), 787–797 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cheng, M.C.N.: K3 Surfaces, N = 4 Dyons, and the Mathieu Group M 24 (May 2010). http://inspirehep.net/record/856699/export/hx and http://inspirehep.net/record/856699/export/hlxe

  11. Cheng, M.C.N., Duncan, J.F.R.: On Rademacher Sums, the Largest Mathieu Group, and the Holographic Modularity of Moonshine (October 2011). http://inspirehep.net/record/940401/export/hx and http://inspirehep.net/record/940401/export/hlxu

  12. Cheng, M.C.N., Duncan, J.F.R.: Mathieu 24 and (Mock) automorphic forms. In: Contribution to AMS Proceedings of Symposia in Pure Mathematics (String-Math 2011). arXiv:1201.4140 [math.RT]

    Google Scholar 

  13. Cheng, M.C.N., Duncan, J.F.R., Harvey, J.: Umbral moonshine. arXiv:1204.2779 [math.RT] (submitted CNTP); Cheng, M.C.N., Duncan, J.F.R., Harvey, J.: Umbral moonshine and the Niemeier lattices. Res. Math. Sci. (Springer) 1, 3 (2014). arXiv:1307.5793 [math.RT]. http://www.resmathsci.com/content/1/1/3

  14. Conway, J.H., Norton, S.P.: Monstrous Moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979)

    MATH  MathSciNet  Google Scholar 

  15. Conway, J., McKay, J., Sebbar, A.: On the discrete groups of Moonshine. Proc. Am. Math. Soc. 132, 2233–2240 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cummins, C.J.: Congruence subgroups of groups commensurable with \(\mathrm{PSL}(2, \mathbb{Z})\) of genus 0 and 1. Exp. Math. 13(3), 361–382 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dabholkar, A., Gomes, J., Murthy, S.: Localization and exact holography. J High Energy Phys. 1304, 062 (2011). doi:10.1007/JHEP04(2013)062

  18. Dabholkar, A., Murthy, S., Zagier, D.: Quantum Black holes, wall-crossing, and mock modular forms (2012). arXiV:1208.4074

    Google Scholar 

  19. de Azevedo Pribitkin, W.: The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight. I. Acta Arith. 91(4), 291–309 (1999)

    MATH  Google Scholar 

  20. de Azevedo Pribitkin, W.: The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight. II. Acta Arith. 93(4), 343–358 (2000)

    MATH  Google Scholar 

  21. de Azevedo Pribitkin, W.: A generalization of the Goldfeld-Sarnak estimate on Selberg’s Kloosterman zeta-function. Forum Math. 12(4), 449–459 (2000)

    MATH  MathSciNet  Google Scholar 

  22. de Boer, J., Cheng, M.C.N., Dijkgraaf, R., Manschot, J., Verlinde, E.: A farey tail for attractor black holes. J. High Energy Phys. 11, 024 (2006)

    Article  Google Scholar 

  23. Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. J. High Energy Phys. 1111, 129, 149 pp. (2011) [21 figures]

    Google Scholar 

  24. Dijkgraaf, R., Maldacena, J.M., Moore, G.W., Verlinde, E.P.: A Black hole Farey tail (2000). hep-th/0005003

    Google Scholar 

  25. Duncan, J.F.R., Frenkel, I.B.: Rademacher sums, moonshine and gravity. Commun. Number Theory Phys. 5(4), 1–128 (2011)

    Article  MathSciNet  Google Scholar 

  26. Eguchi, T., Hikami, K.: Superconformal algebras and mock theta functions 2. Rademacher expansion for K3 surface. Commun. Number Theory Phys. 3, 531–554 (2009)

    Google Scholar 

  27. Eguchi, T., Hikami, K.: Note on twisted elliptic genus of K3 surface. Phys. Lett. B694, 446–455 (2011)

    Article  MathSciNet  Google Scholar 

  28. Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the K3 Surface and the Mathieu group M 24. Exp. Math. 20, 91–96 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Folsom, A., Ono, K.: Duality involving the mock theta function f(q). J. Lond. Math. Soc. (2) 77(2), 320–334 (2008)

    Google Scholar 

  30. Frenkel, I.B., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc. Natl. Acad. Sci. USA 81(10, Phys. Sci.), 3256–3260 (1984)

    Google Scholar 

  31. Frenkel, I.B., Lepowsky, J., Meurman, A.: A moonshine module for the Monster. In: Vertex Operators in Mathematics and Physics (Berkeley, Calif., 1983). Mathematical Sciences Research Institute Publications, vol. 3, pp. 231–273. Springer, New York (1985)

    Google Scholar 

  32. Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Mathematics, vol. 134. Academic, Boston (1988)

    Google Scholar 

  33. Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu Moonshine in the elliptic genus of K3. J. High Energy Phys. 1010, 062 (2010)

    Article  MathSciNet  Google Scholar 

  34. Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu twining characters for K3. J. High Energy Phys. 1009, 058, 19 pp. (2010)

    Google Scholar 

  35. Goldfeld, D., Sarnak, P.: Sums of Kloosterman sums. Invent. Math. 71(2), 243–250 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  36. Guerzhoy, P.: On weak harmonic Maass-modular grids of even integral weights. Math. Res. Lett. 16(1), 59–65 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  37. Knopp, M.I.: Construction of a class of modular functions and forms. Pacific J. Math. 11, 275–293 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  38. Knopp, M.I.: Construction of a class of modular functions and forms. II. Pacific J. Math. 11, 661–678 (1961)

    Article  MathSciNet  Google Scholar 

  39. Knopp, M.I.: Construction of automorphic forms on H-groups and supplementary Fourier series. Trans. Am. Math. Soc. 103, 168–188 (1962)

    MathSciNet  Google Scholar 

  40. Knopp, M.I.: On abelian integrals of the second kind and modular functions. Am. J. Math. 84, 615–628 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  41. Knopp, M.I.: On the Fourier coefficients of small positive powers of θ(τ). Invent. Math. 85(1), 165–183 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  42. Knopp, M.I.: On the Fourier coefficients of cusp forms having small positive weight. In: Theta Functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987). Proceedings of Symposia in Pure Mathematics, vol. 49, pp. 111–127. American Mathematical Society, Providence (1989)

    Google Scholar 

  43. Knopp, M.I.: Rademacher on J(τ), Poincaré series of nonpositive weights and the Eichler cohomology. Not. Am. Math. Soc. 37(4), 385–393 (1990)

    MATH  MathSciNet  Google Scholar 

  44. Knopp, M., Robins, S.: Easy proofs of Riemann’s functional equation for ζ(s) and of Lipschitz summation. Proc. Am. Math. Soc. 129(7), 1915–1922 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kowalski, E.: Poincaré and analytic number theory. In: The Scientific Legacy of Poincaré. History of Mathematics, vol. 36, pp. 73–85. American Mathematical Society, Providence (2010)

    Google Scholar 

  46. Kraus, P., Larsen, F.: Partition functions and elliptic genera from supergravity. J. High Energy Phys. 0701, 002 (2007)

    Article  MathSciNet  Google Scholar 

  47. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998)

    MATH  MathSciNet  Google Scholar 

  48. Maldacena, J., Strominger, A.: Ads3 black holes and a stringy exclusion principle. J. High Energy Phys. 9812, 005 (1998)

    Article  MathSciNet  Google Scholar 

  49. Maloney, A., Witten, E.: Quantum gravity partition functions in three dimensions. J. High Energy Phys. 2, 029, 58 (2010)

    Google Scholar 

  50. Manschot, J., Moore, G.W.: A modern fareytail. Commun. Number Theory Phys. 4, 103–159 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  51. Moore, G.W.: Les Houches Lectures on Strings and Arithmetic (2004). hep-th/0401049

    Google Scholar 

  52. Murthy, S., Nawata, S.: Which AdS3 configurations contribute to the SCFT2 elliptic genus? (2011). arXiV:1112.4844

    Google Scholar 

  53. Murthy, S., Pioline, B.: A Farey tale for N=4 dyons. J. High Energy Phys. 0909, 022 (2009)

    Article  MathSciNet  Google Scholar 

  54. Niebur, D.: Construction of automorphic forms and integrals. Trans. Am. Math. Soc. 191, 373–385 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  55. Petersson, H.: Theorie der automorphen Formen beliebiger reeller Dimension und ihre Darstellung durch eine neue Art Poincaréscher Reihen. Math. Ann. 103(1), 369–436 (1930)

    Article  MATH  MathSciNet  Google Scholar 

  56. Petersson, H.: Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math. 58(1), 169–215 (1932)

    Article  MathSciNet  Google Scholar 

  57. Petersson, H.: Über die Entwicklungskoeffizienten einer allgemeinen Klasse automorpher Formen. Math. Ann. 108(1), 370–377 (1933)

    Article  MathSciNet  Google Scholar 

  58. Poincaré, H.: Fonctions modulaires et fonctions fuchsiennes. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 3, 125–149 (1911)

    Google Scholar 

  59. Rademacher, H.: On the partition function p(n). Proc. Lond. Math. Soc. (2) 43, 241–254 (1937)

    Google Scholar 

  60. Rademacher, H.: The Fourier coefficients of the modular invariant J(τ). Am. J. Math. 60(2), 501–512 (1938)

    Article  MathSciNet  Google Scholar 

  61. Rademacher, H.: The Fourier series and the functional equation of the absolute modular invariant J(τ). Am. J. Math. 61(1), 237–248 (1939)

    Article  MathSciNet  Google Scholar 

  62. Rademacher, H., Zuckerman, H.S.: On the Fourier coefficients of certain modular forms of positive dimension. Ann. Math. (2) 39(2), 433–462 (1938)

    Google Scholar 

  63. Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. Lond. Math. Soc. 16, 315–316 (1917)

    MATH  Google Scholar 

  64. Rouse, J.: Zagier duality for the exponents of Borcherds products for Hilbert modular forms. J. Lond. Math. Soc. (2) 73(2), 339–354 (2006)

    Google Scholar 

  65. Selberg, A.: On the estimation of Fourier coefficients of modular forms. In: Proceedings of Symposia in Pure Mathematics, vol. VIII, pp. 1–15. American Mathematical Society, Providence (1965)

    Google Scholar 

  66. Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan, vol. 11. Iwanami Shoten, Publishers, Tokyo (1971). Kanô Memorial Lectures, No. 1

    Google Scholar 

  67. Sivaramakrishnan, R.: Classical Theory of Arithmetic Functions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 126. Marcel Dekker, New York (1989)

    Google Scholar 

  68. Thompson, J.G.: Finite groups and modular functions. Bull. Lond. Math. Soc. 11(3), 347–351 (1979)

    Article  MATH  Google Scholar 

  69. Thompson, J.G.: Some numerology between the Fischer-Griess Monster and the elliptic modular function. Bull. Lond. Math. Soc. 11(3), 352–353 (1979)

    Article  MATH  Google Scholar 

  70. Zagier, D.: Traces of singular moduli. In: Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998). International Press Lecture Series, vol. 3, pp. 211–244. International Press, Somerville (2002)

    Google Scholar 

  71. Zagier, D.: Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), Séminaire Bourbaki. Vol. 2007/2008. Astérisque 326 (2009), Exp. No. 986, vii–viii, 143–164 (2010)

    Google Scholar 

  72. Zwegers, S.: Mock Theta Functions. Ph.D. thesis, Utrecht University (2002)

    Google Scholar 

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Authors and Affiliations

  1. CNRS and Université Paris 7, UMR CNRS, 7586, Paris, France

    Miranda C. N. Cheng

  2. Department of Mathematics, Case Western Reserve University, Cleveland, OH, 44106, USA

    John F. R. Duncan

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  1. Miranda C. N. Cheng
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Correspondence to Miranda C. N. Cheng .

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  1. University of Heidelberg, Department of Mathematics, Heidelberg, Germany

    Winfried Kohnen

  2. University of Heidelberg, Department of Mathematics, Heidelberg, Germany

    Rainer Weissauer

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Cheng, M.C.N., Duncan, J.F.R. (2014). Rademacher Sums and Rademacher Series. In: Kohnen, W., Weissauer, R. (eds) Conformal Field Theory, Automorphic Forms and Related Topics. Contributions in Mathematical and Computational Sciences, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43831-2_6

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