In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham [Gra01], generalizing arguments of Erdős and Pomerance [EP86] and Monier [Mon80]. These bounds are provided for both Jacobi symbol ±1 cases, providing evidence for the existence of several challenge pseudoprimes.
In this article we propose a geometric description of Arthur packets for
p
p
-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of
p
p
-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur’s main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan’s work, including the prediction that Arthur packets are ABV-packets for
p
p
-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
We improve the unconditional explicit bounds for the error term in the prime counting function ψ(x). In particular, we prove that, for all x > 2, we haveand that, for all log x ≥ 3 000, |ψ(x) − x| < 4.47 • 10 −15 x. This compares to results of Platt & Trudgian (2021) who obtained 4.51 • 10 −13 x. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning π(x) will appear in a follow up work.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.