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Multiplicative Properties of the Number of k-Regular Partitions
In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with k-regular partitions. Extending the generating…
On the Number of Parts of Integer Partitions Lying in Given Residue Classes
Improving upon previous work  on the subject, we use Wright’s Circle Method to derive an asymptotic formula for the number of parts in all partitions of an integer n that are in any given…
Minkowski Length of 3D Lattice Polytopes
- Olivia Beckwith, M. Grimm, Jenya Soprunova, Bradley Weaver
- MathematicsDiscret. Comput. Geom.
- 2 February 2012
This paper gives a polytime algorithm for computing L(P) where P is a 3D lattice polytope, which represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes.
Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles
- Olivia Beckwith, Victor Luo, S. Miller, Karen Shen, N. Triantafillou
- 1 December 2011
It is proved that the limiting signed rescaled spectral measure is the semi-circle and a closed-form expression for the expected value is derived and the asymptotics for the variance for the number of vertices in at least one crossing are determined.
The number of parts in certain residue classes of integer partitions
We use the Circle Method to derive asymptotics for functions related to the number of parts of partitions in particular residue classes.
The Average Gap Distribution for Generalized Zeckendorf Decompositions
An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of…
Generalized Ramanujan Primes
In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x∕2, x]. This was proved by Chebyshev in 1860 and then generalized by Ramanujan in 1919. He showed that…
Indivisibility of class numbers of imaginary quadratic fields
- Olivia Beckwith
- 14 December 2016
We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to $$-X$$-X whose class numbers…
Scarcity of congruences for the partition function
The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell…
Nonholomorphic Ramanujan-type congruences for Hurwitz class numbers
- Olivia Beckwith, Martin Raum, Olav K. Richter
- MathematicsProceedings of the National Academy of Sciences
- 15 April 2020
It is discovered that Ramanujan-type congruences for Hurwitz class numbers can be supported on nonholomorphic generating series and a divisibility result is established for such non holomorphic congruence of Hurwitzclass numbers.