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Barry Brent

Publications and source records attributed to Barry Brent.

13 recordsLinked to original sources
arXiv2026

Coherent sequences of matrix spectra

We attach sequences of matrix spectra to sequences of rational numbers and study them with numerical experiments. The initial segments of the spectrum sequences associated to each of several rational sequences appear to exhibit geometric regularities. Identities originating in the theory of symmetric functions express members of arithmetic sequences $\{h_n\}_{n \ge 0} \thinspace (h_0 = 1)$ as $h_n = |J_{h,n}|/n !$ for particular matrices $J_{h,n}$, where $|\cdot|$ is the determinant. In this setting we studied numerically sequences $\{a(n)\}_{n \ge 1}$ obtained from the Fourier expansions of cusp forms. Within the range of our observations, the sequences of spectra of certain "treated" matrices $\{J^{(c)}_{a,n}\}_{n=1,2,3,...}$ exhibit coherent behavior, meaning that the plots of the minimum moduli among the eigenvalues of the $J^{(c)}_{a,n}$ appear to oscillate or to form approximately straight lines of non-negative slope once $n$ is large enough.

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arXiv2026

Ramanujan's function on small primes

We denote functions mapping n to the Fourier coefficient of q^n in the expansion of a cusp form as Ramanujan functions. We empirically study the eigenvalues of determinants that represent values of these Ramanujan functions. In some cases, considered as point sets in the complex plane, they appear to oscillate as n increases. We look for regularities in this phenomenon and discuss the possibility of exploiting it to attack Lehmer's question about the existence of zeros of Ramanujan's tau function.

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arXiv2025

Matrix methods for arithmetic functions

We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.

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arXiv2023

Finite field models of Raleigh-Akiyama polynomials for Hecke groups

Following work of Raleigh and Akiyama (\cite{raleigh1962fourier, akiyama1992note}), in \cite{interpolating} we considered (among other objects) families of weight zero meromorphic modular forms $J_m$ for Hecke groups $G(λ_m)$. We conjectured in \cite{interpolating} that, for a certain uniformizing variable $X_m$, the $J_m$ have Fourier expansions $J_m = 1/X_m + \sum_{n = 0}^{\infty} A_n(m) X_m^n$, where the $A_n(x)$ are polynomials in $\mathbb{Q}[x]$. The present article is concerned with models $\mathcal{A}_n[p](x)$ of the $A_n(x)$: polynomials representing self-maps of finite fields with characteristic $p$. The main content is a conjecture specifying $\mathcal{A}_n[p](x)$ up to a multiplicative constant for certain families of $n$ and $p$, based on numerical experiments.

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arXiv2022

On the constant terms of certain meromorphic modular forms for Hecke groups

We study polynomials interpolating the (rational) constant terms of certain meromorphic modular forms for Hecke groups. We make observations about the divisibility properties of the constant terms and connect them to several sequences, for example, to O.E.I.S. sequence A005148 \cite{OEISNewmanShanks}, which was studied by Newman, Shanks and Zagier \cite{newman2004sequence}, \cite{newman2004sequenceAppendix} in an article on its use in series approximations to $π$.

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arXiv2021

Polynomial interpolation of modular forms for Hecke groups

Extending work of J. Raleigh, we compute polynomials $P_{n,F}(x)$ associated to certain families $F = \{f_m\}_{m = 3, 4, ...}$ of modular forms for Hecke groups $G(λ_m)$ with the property that $P_{n,F}(m)$ is the $n^{th}$ coefficient in the Fourier expansion of $f_m$. We express the $P_{n,F}$ in terms of the Fourier expansions of well-known Hauptmoduln, or in terms of certain divisor-sums. By studying the complex roots of the $P_n$, we relate them to Lehmer's question about Ramanujan's tau function. We review the theory of triangle functions and Hecke's theory of modular forms in order to establish a basis for our code, some of which originates in the dissertation of J. Leo. The article is an account of numerical experiments; the only theorems in it belong to work by others that we review as described above.

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arXiv2021

Hecke groups, linear recurrences, and Kepler limits

We study the linear fractional transformations in the Hecke group $G(Φ)$ where $Φ$ is either root of $x^2 - x -1$ (the larger root being the "golden ratio" $ϕ= 2 \cos \frac π5$.) Let $g \in G(Φ)$ and let $z$ be a generic element of the upper half-plane. Exploiting the fact that $Φ^2 = Φ-1$, we find that $g(z)$ is a quotient of linear polynomials in $z$ such that the coefficients of $z^1$ and $z^0$ in the numerator and denominator of $g(z)$ appear themselves to be linear polynomials in $Φ$ with coefficients that are certain multiples of Fibonacci numbers. We make somewhat less detailed observations along similar lines about the functions in $G(2 \cos \frac πk)$ for $k \geq 5$.

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arXiv2017

Variants of the Riemann zeta function

We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically, we study the family of functions $V_z: s \mapsto ζ(s) \exp (zs)$. We observe convergence of $V_z$ fixed points along nearly logarithmic spirals with initial points at zeta fixed points and centered upon Riemann zeros. We can approximate these spirals numerically, so they might afford a means to study the geometry of the relationship of zeta fixed points to Riemann zeros.

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arXiv2017

Experiments with the dynamics of the Riemann zeta function

We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero $ρ$ (trivial or nontrivial) and each zeta fixed point $ψ$ there is a nearly logarithmic spiral $s_{ρ, ψ}$ with center $ψ$ containing $ρ$. (2) $s_{ρ, ψ}$ interpolates a subset $B_{ρ, ψ}$ of the backward zeta orbit of $ρ$ comprising a set of zeros of all iterates of zeta. (3) If zeta is viewed as a function on sets, $ζ(B_{ρ, ψ}) = B_{ρ, ψ} \cup \{0 \}$. (4) $B_{ρ, ψ}$ has nearly uniform angular distribution around the center of $s_{ρ, ψ}$. We will make these statements precise.

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arXiv2002

3x+1 dynamics on rationals with fixed denominator

We propose the existence of an infinite class of exact analogues of the 3x+1 conjecture for rational numbers with fixed denominators. For some other denominators, there are several attracting cycles, which exhibit scaling and covariance phenomena. We analyze these phenomena in terms of results of Bohm, Sontacchi and Lagarias.

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arXiv2000

Quadratic minima and modular forms II

We give upper bounds on the size of the gap between a non-zero constant term and the next non-zero Fourier coefficient of an entire level two modular form. We give upper bounds for the minimum positive integer represented by a level two even positive-definite quadratic form. These bounds extend partial results in part I.

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arXiv1998

Quadratic minima and modular forms

We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for Γ_0(2). Numerical evidence indicates that a sharper bound holds for the weights h \equiv 2 . We derive upper bounds for the minimum positive integer represented by level two even positive-definite quadratic forms. Our data suggest that, for certain meromorphic modular forms and p=2,3, the p-order of the constant term is related to the base-p expansion of the order of the pole at infinity, and they suggest a connection between divisibility properties of the Ramanujan tau function and those of the Fourier coefficients of 1/j.

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