A derivation of the Hardy-Ramanujan formula from an arithmetic formula
HTML articles powered by AMS MathViewer
- by Michael Dewar and M. Ram Murty PDF
- Proc. Amer. Math. Soc. 141 (2013), 1903-1911 Request permission
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
- J. H. Bruinier and K. Ono. Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms. arXiv:1104.1182, Apr. 2011.
- 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, DOI 10.1007/s00208-008-0252-1
- J. H. Bruinier, K. Ono, and A. Sutherland. Class polynomials for nonholomorphic modular functions, in preparation.
- Duncan A. Buell, Binary quadratic forms, Springer-Verlag, New York, 1989. Classical theory and modern computations. MR 1012948, DOI 10.1007/978-1-4612-4542-1
- 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, DOI 10.1007/BF01458081
- G. H. Hardy and S. Ramanujan. Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2), 17:75–115, 1918.
- H. Rademacher. On the partition function $p(n)$. Proc. London Math. Soc. (2), 43:241–254, 1937.
- Atle Selberg, Collected papers. Vol. I, Springer-Verlag, Berlin, 1989. With a foreword by K. Chandrasekharan. MR 1117906
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
- MR Author ID: 128555
- Email: murty@mast.queensu.ca
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3034417