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Nathan C. Ryan

Publications and source records attributed to Nathan C. Ryan.

At least 19 recordsLinked to original sources
arXiv2024

Repulsion of zeros close to $s=1/2$ for L-functions

In this paper we present results of several experiments in which we model the repulsion of low-lying zeros of L-functions using random matrix theory. Previous work has typically focused on the twists of L-functions associated to elliptic curves and on families that can be modeled by unitary and orthogonal matrices. We consider families of L-function of modular forms of weight greater than 2 and we consider families that can be modeled by symplectic matrices. Additionally, we explore a model for low-lying zeros of twists that incorporates a discretization arising from the Kohnen--Zagier theorem. Overall, our numeric evidence supports the expectation that the repulsion of zeros decreases as the conductor of the twist increases. Surprisingly, though, it appears that using the discretization that arises from the Kohnen--Zagier theorem does not model the data better than if the discretization is not used for forms of weight 4 or above.

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arXiv2023

Explicit families of congruences for the overpartition function

In this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo $m$ for $m = 5, 7, 11$, and finite families for $m = 13, 17, 19$.

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arXiv2023

Efficient computation of the overpartition function and applications

In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula, and we use this method to find many new Ramanujan-style congruences, whose existence is predicted by Treneer and a few of which were first discovered by Ryan, Scherr, Sirolli and Treneer.

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arXiv2023

Congruences satisfied by eta-quotients

The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general Fourier coefficients state the existence of infinitely many families of congruences. In this article we give an algorithm for computing explicit instances of such congruences for eta-quotients. We illustrate our method with a few examples.

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arXiv2023

Vanishing of Quartic and Sextic Twists of $L$-functions

Let $E$ be an elliptic curve over $\mathbf{Q}$. We conjecture asymptotic estimates for the number of vanishings of $L(E,1,χ)$ as $χ$ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on $E$. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Earlier work by David, Fearnley and Kisilevsky formulates analogous conjectures for characters of any odd prime order. In the composite order case, however, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. Along the way we introduce the notion of totally order $\ell$ characters to quantify how quickly quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (sextic, resp.) characters is much slower than in the sub-family of totally quartic (sextic, resp.) characters. A conceptual explanation for this phenomenon is that the full family of order $\ell$ twisted elliptic curve $L$-functions, with $\ell$ even and composite, is a mixed family with both unitary and orthogonal aspects.

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arXiv2021

Analysis of the Pennsylvania Additive Classification Tool: Biases and Important Features

The Pennsylvania Additive Classification Tool (PACT) is a carceral algorithm used by the Pennsylvania Department of Corrections in order to determine the security level for an incarcerated person in the state's prison system. For a newly incarcerated person it is used in their initial classification. The initial classification can be overridden both for discretionary and administrative reasons. An incarcerated person is reclassified annually using a variant of the PACT and this reclassification can be overridden, too, and for similar reasons. In this paper, for each of these four processes (the two classifications and their corresponding overrides), we develop several logistic models, both binary and multinomial, to replicate these processes with high accuracy. By examining these models, we both identify which features are most important in the model and quantify and describe biases that exist in the PACT, its overrides, and its use in reclassification. Because the details of how the PACT operates have been redacted from public documents, it is important to know how it works and what disparate impact it might have on different incarcerated people.

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arXiv2021

Analyzing a Carceral Algorithm used by the Pennsylvania Department of Corrections

Scholars have focused on algorithms used during sentencing, bail, and parole, but little work explores what we call carceral algorithms that are used during incarceration. This paper is focused on the Pennsylvania Additive Classification Tool (PACT) used to classify prisoners' custody levels while they are incarcerated. Algorithms that are used during incarceration warrant deeper attention by scholars because they have the power to enact the lived reality of the prisoner. The algorithm in this case determines the likelihood a person would endure additional disciplinary actions, can complete required programming, and gain experiences that, among other things, are distilled into variables feeding into the parole algorithm. Given such power, examining algorithms used on people currently incarcerated offers a unique analytic view to think about the dialectic relationship between data and algorithms. Our examination of the PACT is two-fold and complementary. First, our qualitative overview of the historical context surrounding PACT reveals that it is designed to prioritize incapacitation and control over rehabilitation. While it closely informs prisoner rehabilitation plans and parole considerations, it is rooted in population management for prison securitization. Second, on analyzing data for 146,793 incarcerated people in PA, along with associated metadata related to the PACT, we find it is replete with racial bias as well as errors, omissions, and inaccuracies. Our findings to date further caution against data-driven criminal justice reforms that rely on pre-existing data infrastructures and expansive, uncritical, data-collection routines.

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arXiv2021

Uncertainty in Criminal Justice Algorithms: simulation studies of the Pennsylvania Additive Classification Tool

Much attention has been paid to algorithms related to sentencing, the setting of bail, parole decisions and recidivism while less attention has been paid to carceral algorithms, those algorithms used to determine an incarcerated individual's lived experience. In this paper we study one such algorithm, the Pennsylvania Additive Classification Tool (PACT) that assigns custody levels to incarcerated individuals. We analyze the PACT in ways that criminal justice algorithms are often analyzed: namely, we train an accurate machine learning model for the PACT; we study its fairness across sex, age and race; and we determine which features are most important. In addition to these conventional computations, we propose and carry out some new ways to study such algorithms. Instead of focusing on the outcomes themselves, we propose shifting our attention to the variability in the outcomes, especially because many carceral algorithms are used repeatedly and there can be a propagation of uncertainty. By carrying out several simulations of assigning custody levels, we shine light on problematic aspects of tools like the PACT.

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arXiv2019

Analytic evaluation of Hecke eigenvalues for classical modular forms

We propose a method for computing approximations to the Hecke eigenvalues of a classical modular eigenform $f$, based on the analytic evaluation of $f$ at points in the upper half plane. Our approach works with arbitrary precision, allows for a strict control of the error in the approximation, and outperforms current exact computation methods.

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arXiv2019

Theta series and number fields: theorems and experiments

We construct certain $θ$-series associated to number fields and prove that for number fields of degree less than equal to 4, these $θ$-series are number field invariants. We also investigate whether or not the collection of $θ$-series associated to number fields of the same degree and discriminant are linearly independent. This is known to be true if the degree of the number field is less than or equal to 3. We do not prove in this paper that they are linearly independent in general but we do give computational and heuristic evidence that we would expect them to be.

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arXiv2018

Analytic evaluation of Hecke eigenvalues for Siegel modular forms of degree two

The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform F is to determine a large number of Fourier coefficients of F and then compute the Hecke action on those coefficients. We present a new method based on the numerical evaluation of F at explicit points in the upper-half space and of its image under the Hecke operators. The approach is more efficient than the standard method and has the potential for further optimization by identifying good candidates for the points of evaluation, or finding ways of lowering the truncation bound. A limitation of the algorithm is that it returns floating point numbers for the eigenvalues; however, the working precision can be adjusted at will to yield as close an approximation as needed.

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arXiv2017

Analytic $L$-functions: Definitions, Theorems, and Connections

$L$-functions can be viewed axiomatically, such as in the formulation due to Selberg, or they can be seen as arising from cuspidal automorphic representations of $\textrm{GL}(n)$, as first described by Langlands. Conjecturally these two descriptions of $L$-functions are the same, but it is not even clear that these are describing the same set of objects. We propose a collection of axioms that bridges the gap between the very general analytic axioms due to Selberg and the very particular and algebraic construction due to Langlands. Along the way we prove theorems about $L$-functions that satisfy our axioms and state conjectures that arise naturally from our axioms.

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arXiv2016

Computing Jacobi Forms

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows to generate a Jacobi eigenform directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

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arXiv2013

Evaluating $L$-functions with few known coefficients

We address the problem of evaluating an $L$-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that is possible to evaluate the $L$-function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.

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arXiv2013

Characterizations of the Saito-Kurokawa lifting: a survey

There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree 2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations of Saito-Kurokawa lifts.

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arXiv2013

Multiplicity one for $L$-functions and applications

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to $L$-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for $L$-functions. Significantly stronger results are available when the L-function is known to be automorphic.

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arXiv2012

Computations of vector-valued Siegel modular forms

We carry out some computations of vector valued Siegel modular forms of degree two, weight (k,2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results: (1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms and Siegel modular forms.

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