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Andrew Fiori

Publications and source records attributed to Andrew Fiori.

At least 19 recordsLinked to original sources
arXiv2026

Bounds for Mertens Sums

In this article we provide new bounds for the Mertens sums and products including $\sum_{p \le x} p^{-1}$ and $\prod_{p \le x} (1-\frac{1}{p})$ which provide superior exponential and log type bounds for these sums in all ranges. These weighted prime number sums and products were extensively studied by Rosser and Schoenfeld (1962) and are employed in a wide range of applications in number theory, cryptography, and combinatorics. Extensive tables are provided in this article which will be useful for these types of applications. The main new ideas in this article are sharp bounds for weighted sums of zeros of zeros of the zeta function. We make use of a novel technique of Fiori-Kadiri-Swidinsky (2023) which relies on a recent explicit zero-density estimate for $N(\sigma,T)$ of Kadiri-Lumley-Ng (2018). The bounds and techniques in this article for weighted zeros sums will likely be useful in many other arithmetic applications. Our main theorem significantly improves the exponential decay result of Vanlalngaia (2017) and fills a gap in the literature by correcting work of Dusart (2018). The results are also presented in a way that are amenable to future improvements. In addition, we prove an exact ``Riemann-Guinand explicit formula" for the Mertens sum $\sum_{p \le x} p^{-1}$ that appears to be new.

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arXiv2025

Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters

Let $G$ be a reductive $p$-adic group for which the local Langlands correspondence is known, and $\lambda$ an infinitesimal parameter of $G$. In this paper, we prove that if the $p$-adic Kazhdan-Lusztig hypothesis holds for $\lambda$, then for a Langlands parameter $\phi$ with infinitesimal parameter $\lambda$, if $\Pi_\phi(G)$ contains a generic representation, then $L(s, \phi, \Ad)$ is regular at $s=1$. We then prove an analogous statement for ABV-packets, which together with Vogan's conjecture on ABV-packets implies that if Arthur's conjectures for $G$ are known, then one direction of Shahidi's enhanced genericity conjecture holds: If an Arthur packet $\Pi_\psi(G)$ contains a generic representation, then $\phi_\psi$ is tempered. We also offer some speculation about the relationship between Arthur type representations and singularities in varieties of Langlands parameters defined by Vogan.

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arXiv2025

Counting Frobenius Pseudoprimes

We generalize the work of Erdos-Pomerance and Fiori-Shallue on counting Frobenius pseudoprimes from the cases of degree one and two respectively to arbitrary degree. More specifically we provide formulas for counting the number of false witnesses for a number $n$ with respect to Grantham's Frobenius primality test. We also provide conditional assymptotic lower bounds on the average number of Frobenius pseudoprimes and assymptotic upper bounds on the same.

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arXiv2025

A Note on the Phragmen-Lindelof Theorem

We provide a generalization of the Phragm\'en-Lindel\"of principal of Rademacher with the aim of correcting, or at least provide a pathway to correcting, several errors appearing in the literature.

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arXiv2025

The moduli space of representations of the modular group into $G_2$

In this paper we construct a large four-dimensional family of representations of the modular group into $G_2$. Precisely, this family is an etale cover of degree $96$ of an open subset of the moduli space of such representations. This moduli space has two main components, of dimensions one and four. The one-dimensional component consists of well-studied rigid representations, in the sense of Katz. We focus on the four-dimensional component which consists of representations that are not rigid. We also provide algebraic conditions to ensure that the specializations surject onto $G_2(\mathbf{F}_p)$ for primes $p\geq 5$. These representations give new examples of $\phi$-congruence subgroups of the modular group as introduced in previous work.

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arXiv2024

Whittaker normalization of $p$-adic ABV-packets and Vogan's conjecture for tempered representations

We show that ABV-packets for $p$-adic groups do not depend on the choice of a Whittaker datum, but the function from the ABV-packet to representations of the appropriate microlocal equivariant fundamental group does, and we find this dependence exactly. We study the relation between open parameters and tempered parameters and Arthur parameters and generic representations. We state a genericity conjecture for ABV-packets and prove this conjecture for quasi-split classical groups and their pure inner forms. Motivated by this we study ABV-packets for open parameters and prove that they are L-packets, and further that the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by the Langlands correspondence. From this conclude Vogan's conjecture on A-packets for tempered representations: ABV-packets for tempered parameters are Arthur packets and the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by Arthur.

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arXiv2024

Generic representations, open parameters and ABV-packets for $p$-adic groups

If $\pi$ is a representation of a $p$-adic group $G(F)$, and $\phi$ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of $\phi$ that will detect when $\pi$ is generic? In this paper we show that if $G$ is classical or if we assume the Kazhdan-Lusztig hypothesis for $G$, then the answer is yes, and the property is that the orbit of $\phi$ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter $\phi$ for a $p$-adic group $G(F)$, the ABV-packet $\Pi^{\mathrm{ABV}}_\phi(G(F))$ contains a generic representation if and only if the local adjoint L-function $L(s,\phi,\mathop{\text{Ad}})$ is regular at $s=1$, and show that this condition is equivalent to the "open parameter" condition above. We show that this genericity conjecture for ABV-packets follows from other standard conjectures and we verify its validity with the same conditions on $G$. We show that, in this case, the ABV-packet for $\phi$ coincides with its $L$-packet. Finally, we prove Vogan's conjecture on $A$-packets for tempered parameters.

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arXiv2024

The Least Prime in Arithmetic an Progression

In his 1979 paper Samuel Wagstaff studied the problem of bounding the first prime in an arithmetic progression. In this paper we update a number of his computations using advances in hardware. Based on this we refine his conjecture on Primes in Arithmetic Progression and provide further numerical evidence in support of it. For instance we conjecture that for $n>3$ we have a bound of $3\phi(n)\log(n)\log(\phi(n))$ and verified this bound to $10^8$.

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arXiv2022

Families of $ϕ$-congruence subgroups of the modular group

We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed $ϕ$-congruence subgroups, are obtained by reducing homomorphisms $ϕ$ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi-unipotent case we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve $y^2=x^3-1728$ defined by the commutator subgroup of the modular group.

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arXiv2022

Sharper bounds for the error term in the Prime Number Theorem

We provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function $\psi(x)$ to bounds of the same type on both $\theta(x)$ and $\pi(x)$. This follows up our previous work on $\psi(x)$ in \cite{FKS}, and prove that $ | \pi(x) - \mathrm{Li}(x) | \leq 9.2211\, x\sqrt{\log(x)} \exp \big( -0.8476 \sqrt{\log(x)} \big) $ for all $x\ge 2$. Additionally, we are able to obtain the best numeric bounds for $x$ on a very large interval (all $x$ up to $\exp(1.8\cdot10^9)$).

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arXiv2022

Sharper bounds for the Chebyshev function $\psi(x)$

We improve the unconditional explicit bounds for the error term in the prime counting function $\psi(x)$. In particular, we prove that, for all $x>2$, we have \[ \left| \psi(x)-x \right| < 9.22106 \, x \, (\log x)^{3/2} \exp(-0.8476836\sqrt{\log x}), \] and that, for all $\log x \ge 3\,000$, \[ \left| \psi(x)-x \right| < 4.47\cdot 10^{-15} x. \] This compares to results of Platt \& Trudgian (2021) who obtained $4.51\cdot 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 $\pi(x)$ will appear in a follow up work.

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arXiv2021

Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.

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arXiv2021

Appearance of the Kashiwara-Saito singularity in the representation theory of $p$-adic $\mathop{GL}_{16}$

In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packet, as defined in earlier work, contains exactly one other representation; remarkably, this other admissible representation is of Arthur type. In the course of this study we strengthen the main result concerning the Kashiwara-Saito singularity. The irreducible admissible representations in this paper illustrate a fact we found surprising: for general linear groups, while all A-packets are singletons, some ABV-packets are not.

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arXiv2021

Arthur packets for $p$-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

In this article we propose a geometric description of Arthur packets for $p$-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 $p$-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, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

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arXiv2021

Toward the endoscopic classification of unipotent representations of $p$-adic $G_2$

We begin this paper by reviewing the Langlands correspondence for unipotent representations of the exceptional group of type $G_2$ over a $p$-adic field $F$ and present it in an explicit form. Then we compute all ABV-packets, as defined in [CFM+21] following ideas from Vogan's 1993 paper The local Langlands Conjecture, and prove that these packets satisfy properties derived from the expectation that they are generalized A-packets. We attach distributions to ABV-packets for $G_2$ and its endoscopic groups and study a geometric endoscopic transfer of these distributions. This paper builds on earlier work by the same authors.

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arXiv2020

Arthur packets for $G_2$ and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the $p$-adic exceptional group $G_2$. Here we treat the most interesting case by defining and computing Arthur packets with component group $S_3$. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classification and strengthen our result on stability.

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arXiv2020

Average liar count for degree-2 Frobenius pseudoprimes

In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham, generalizing arguments of Erdős and Pomerance, and Monier. These bounds are provided for both Jacobi symbol plus and minus cases, providing evidence for the existence of several challenge pseudoprimes.

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