Abstract: In 2005, E. Noel and G. Panos constructed a formula to count the number of semiprimes less than a given value x. In 2006, this formula was rediscovered independently by R. G. Wilson V. However, each citation of this result was simply through personal communication, and a formal proof has not been written down anywhere in the literature. In this article, we generalize their formula to the case of k-almost primes and square-free k-almost primes.
https://arxiv.org/abs/2310.14989
Abstract: In this paper, we prove a converse theorem for half-integral weight modular forms assuming functional equations for L-series with additive twists. This result is an extension of Booker, Farmer, and Lee's result in [BFL22] to the half-integral weight setting. Similar to their work, the main result of this paper is obtained as a consequence of the half-integral weight Petersson trace formula.
https://arxiv.org/abs/2306.02872
Abstract:We show that the category of pastures has arbitrary limits and colimits of diagrams indexed by a small category.
https://arxiv.org/abs/2103.08655
Abstract: We use Young tableaux to compute the dimension of Vr, the Prym-Brill-Noether locus of a folded chain of loops of any gonality. This tropical result yields a new upper bound on the dimensions of algebraic Prym-Brill-Noether loci. Moreover, we prove that Vr is pure-dimensional and connected in codimension 1 when dimVr≥1. We then compute the first Betti number of this locus for even gonality when the dimension is exactly 1, and compute the cardinality when the locus is finite and the edge lengths are generic.
International Mathematics Research Notices , Volume 2022, Issue 4, February 2022
Preprint:https://arxiv.org/abs/1912.02863
Abstract:We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong hyperfield extensions. For quotient hyperfields, we develop a method to construct strong hyperfield extensions that contain roots to any polynomial over the hyperfield. Furthermore, we give an example of a hyperfield that has two non-isomorphic minimal extensions containing a root to some polynomial. This shows that the process of adjoining a root to a hyperfield is not a well-defined operation.
https://arxiv.org/abs/1912.05919
Watch me talk about this paper here