Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

A derivation of the Hardy-Ramanujan formula from an arithmetic formula


Authors: Michael Dewar and M. Ram Murty
Journal: Proc. Amer. Math. Soc. 141 (2013), 1903-1911
MSC (2010): Primary 11E16, 11F03, 11P82
DOI: https://doi.org/10.1090/S0002-9939-2012-11458-3
Published electronically: December 26, 2012
MathSciNet review: 3034417
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We re-prove the Hardy-Ramanujan asymptotic formula for the partition function without using the circle method. We derive our result from recent work of Bruinier and Ono on harmonic weak Maass forms.


References [Enhancements On Off] (What's this?)

  • 1. J. H. Bruinier and K. Ono.
    Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms.
    arXiv:1104.1182, Apr. 2011.
  • 2. Jan H. Bruinier, Ken Ono, and Robert C. Rhoades, Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), no. 3, 673–693. MR 2430995, https://doi.org/10.1007/s00208-008-0252-1
  • 3. J. H. Bruinier, K. Ono, and A. Sutherland.
    Class polynomials for nonholomorphic modular functions,
    in preparation.
  • 4. Duncan A. Buell, Binary quadratic forms, Springer-Verlag, New York, 1989. Classical theory and modern computations. MR 1012948
  • 5. B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of 𝐿-series. II, Math. Ann. 278 (1987), no. 1-4, 497–562. MR 909238, https://doi.org/10.1007/BF01458081
  • 6. G. H. Hardy and S. Ramanujan.
    Asymptotic formulae in combinatory analysis.
    Proc. London Math. Soc. (2), 17:75-115, 1918.
  • 7. H. Rademacher.
    On the partition function $ p(n)$.
    Proc. London Math. Soc. (2), 43:241-254, 1937.
  • 8. Atle Selberg, Collected papers. Vol. I, Springer-Verlag, Berlin, 1989. With a foreword by K. Chandrasekharan. MR 1117906

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11E16, 11F03, 11P82

Retrieve articles in all journals with MSC (2010): 11E16, 11F03, 11P82


Additional Information

Michael Dewar
Affiliation: Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
Email: mdewar@mast.queensu.ca

M. Ram Murty
Affiliation: Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
Email: murty@mast.queensu.ca

DOI: https://doi.org/10.1090/S0002-9939-2012-11458-3
Received by editor(s): September 22, 2011
Published electronically: December 26, 2012
Additional Notes: The first author was supported by a postdoctoral fellowship from the Natural Sciences and Engineering Council of Canada (NSERC)
The second author was supported by a Discovery Grant from NSERC
Communicated by: Ken Ono
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.