## Benford’s law for coefficients of modular forms and partition functions

HTML articles powered by AMS MathViewer

- by Theresa C. Anderson, Larry Rolen and Ruth Stoehr PDF
- Proc. Amer. Math. Soc.
**139**(2011), 1533-1541 Request permission

## Abstract:

Here we prove that Benford’s law holds for coefficients of an infinite class of modular forms. Expanding the work of Bringmann and Ono on exact formulas for harmonic Maass forms, we derive the necessary asymptotics. This implies that the unrestricted partition function $p(n)$, as well as other natural partition functions, satisfies Benford’s law.## References

- F. Benford.
*The law of anomalous numbers.*Proceedings of the American Philosophical Society, 1938, Vol. 78, No. 4, pages 551-572. - Kathrin Bringmann and Ken Ono,
*An arithmetic formula for the partition function*, Proc. Amer. Math. Soc.**135**(2007), no. 11, 3507–3514. MR**2336564**, DOI 10.1090/S0002-9939-07-08883-1 - K. Bringmann and K. Ono.
*Coefficients of harmonic Maass forms.*Proceedings of the 2008 University of Florida Conference on Partitions, $q$-series and Modular Forms, in press. - Persi Diaconis,
*The distribution of leading digits and uniform distribution $\textrm {mod}$ $1$*, Ann. Probability**5**(1977), no. 1, 72–81. MR**422186**, DOI 10.1214/aop/1176995891 - G. H. Hardy and S. Ramanujan,
*Une formule asymptotique pour le nombre des partitions de $n$ [Comptes Rendus, 2 Jan. 1917]*, Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 239–241 (French). MR**2280874** - T. Hill.
*The first-digit phenomenon*, American Scientists**86**(1996), 358-363. - Theodore P. Hill,
*A statistical derivation of the significant-digit law*, Statist. Sci.**10**(1995), no. 4, 354–363. MR**1421567** - Alex V. Kontorovich and Steven J. Miller,
*Benford’s law, values of $L$-functions and the $3x+1$ problem*, Acta Arith.**120**(2005), no. 3, 269–297. MR**2188844**, DOI 10.4064/aa120-3-4 - L. Kuipers and H. Niederreiter,
*Uniform distribution of sequences*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR**0419394** - Simon Newcomb,
*Note on the Frequency of Use of the Different Digits in Natural Numbers*, Amer. J. Math.**4**(1881), no. 1-4, 39–40. MR**1505286**, DOI 10.2307/2369148 - Ken Ono,
*The web of modularity: arithmetic of the coefficients of modular forms and $q$-series*, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR**2020489** - Sinai Robins,
*Generalized Dedekind $\eta$-products*, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 119–128. MR**1284055**, DOI 10.1090/conm/166/01645 - Don Zagier,
*Elliptic modular forms and their applications*, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 1–103. MR**2409678**, DOI 10.1007/978-3-540-74119-0_{1}

## Additional Information

**Theresa C. Anderson**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Email: tcanderson@math.brown.edu
**Larry Rolen**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 923990
- ORCID: 0000-0001-8671-8117
- Email: lrolen@wisc.edu
**Ruth Stoehr**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Email: rstoehr@emory.edu
- Received by editor(s): April 30, 2010
- Received by editor(s) in revised form: May 5, 2010
- Published electronically: October 5, 2010
- Communicated by: Matthew A. Papanikolas
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 1533-1541 - MSC (2010): Primary 11F12, 11F20, 11P82
- DOI: https://doi.org/10.1090/S0002-9939-2010-10577-4
- MathSciNet review: 2763743