Explicit bounds for the number of $p$-core partitions
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- by Byungchan Kim and Jeremy Rouse PDF
- Trans. Amer. Math. Soc. 366 (2014), 875-902 Request permission
Abstract:
In this article, we derive explicit bounds on $c_{t} (n)$, the number of $t$-core partitions of $n$. In the case when $t = p$ is prime, we express the generating function $f(z)$ as the sum \[ f(z) = e_{p} E(z) + \sum _{i} r_{i} g_{i}(z) \] of an Eisenstein series and a sum of normalized Hecke eigenforms. We combine the Hardy-Littlewood circle method with properties of the adjoint square lifting from automorphic forms on $\rm {GL}(2)$ to $\rm {GL}(3)$ to bound $R(p) := \sum _{i} |r_{i}|$, solving a problem raised by Granville and Ono.
In the case of general $t$, we use a combination of techniques to bound $c_{t}(n)$ and as an application prove that for all $n \geq 0$, $n \ne t+1$, \[ c_{t+1}(n) \geq c_{t}(n) \] provided $4 \leq t \leq 198$, as conjectured by Stanton.
References
Additional Information
- Byungchan Kim
- Affiliation: School of Liberal Arts, Seoul National University of Science and Technology, 172 Gongreung 2 dong, Nowongu, Seoul,139-743, Korea
- MR Author ID: 847992
- Email: bkim4@seoultech.ac.kr
- Jeremy Rouse
- Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
- MR Author ID: 741123
- Email: rouseja@wfu.edu
- Received by editor(s): November 30, 2009
- Received by editor(s) in revised form: May 22, 2012
- Published electronically: August 19, 2013
- Additional Notes: The second author was supported by NSF grant DMS-0901090
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 875-902
- MSC (2010): Primary 11P82; Secondary 11P55, 11F66
- DOI: https://doi.org/10.1090/S0002-9947-2013-05883-7
- MathSciNet review: 3130320