149 citations · 319 across the 56 of their papers we have counts for
67 papers
Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation
Stephan Ramon Garcia
We study algebraic relations among functions in star-invariant subspaces. Let be an inner function and suppose that satisfy , in which is a hom…
Fermat's Last Theorem in star-invariant subspaces: capacity-zero boundary spectrum
Stephan Ramon Garcia
We prove that the Fermat equation with exponent has no nontrivial solutions in the star-invariant subspace , in which , wh…
Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms
Ángel Chávez, Stephan Ramon Garcia, Jackson Hurley
We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily…
Symmetric tensor powers of graphs
Weymar Astaiza, Alexander J. Barrios, Henry Chimal-Dzul +5
The symmetric tensor power of graphs is introduced and its fundamental properties are explored. A wide range of intriguing phenomena occur when one considers symmetric tensor power…
Symmetric and Antisymmetric Tensor Products for the Function-Theoretic Operator Theorist
Stephan Ramon Garcia, Ryan O'Loughlin, Jiahui Yu
We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator th…
The Reciprocal Schur Inequality
Albrecht Boettcher, Stephan Ramon Garcia, Mishko Mitkovski
Schur's inequality states that the sum of three special terms is always nonnegative. This note is a short review of inequalities for the sum of the reciprocals of these terms and o…
The strongly Leibniz property and the Gromov--Hausdorff propinquity
Konrad Aguilar, Stephan Ramon Garcia, Elena Kim +1
We construct a new version of the dual Gromov--Hausdorff propinquity that is sensitive to the strongly Leibniz property. In particular, this new distance is complete on the class o…
The error term in the truncated Perron formula for the logarithm of an -function
Stephan Ramon Garcia, Jeffrey Lagarias, Ethan Simpson Lee
We improve upon the traditional error term in the truncated Perron formula for the logarithm of an -function. All our constants are explicit.
Moments of Gaussian Periods and Modified Fermat Curves
Stephan Ramon Garcia, Brian Lorenz, George Todd
We use supercharacter theory to study moments of Gaussian periods. For and fixed , we compute the fourth absolute moments for all but finitely many primes . For …
Frobenius-Rieffel norms on finite-dimensional C*-algebras
Konrad Aguilar, Stephan Ramon Garcia, Elena Kim
In 2014, Rieffel introduced norms on certain unital C*-algebras built from conditional expectations onto unital C*-subalgebras. We begin by showing that these norms generalize the…
Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups
Stephan Ramon Garcia, Christopher O'Neill, Gabe Udell
For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factoriza…
Norms on complex matrices induced by complete homogeneous symmetric polynomials
Konrad Aguilar, Ángel Chávez, Stephan Ramon Garcia +1
We introduce a remarkable new family of norms on the space of complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their c…
Explicit estimates for Artin -functions: Duke's short-sum theorem and Dedekind Zeta Residues
Stephan Ramon Garcia, Ethan Simpson Lee
Under GRH, we establish a version of Duke's short-sum theorem for entire Artin -functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerica…
A Gallery of Gaussian Periods
Ellen Eischen, Stephan Ramon Garcia
Gaussian periods are certain sums of roots of unity whose study dates back to Gauss's seminal work in algebra and number theory. Recently, large scale plots of Gaussian periods hav…
An effective analytic formula for the number of distinct irreducible factors of a polynomial
Stephan Ramon Garcia, Ethan Simpson Lee, Josh Suh +1
We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial . We use an explicit version o…
Unconditional Explicit Mertens' Theorems for Number Fields and Dedekind Zeta Residue Bounds
Stephan Ramon Garcia, Ethan Simpson Lee
We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for . Our error terms are explicitly bounde…
Explicit Mertens' Theorems for Number Fields
Stephan Ramon Garcia, Ethan Simpson Lee
Assuming the Generalized Riemann Hypothesis we obtain uniform, effective number-field analogues of Mertens' theorems.
Factorization length distribution for affine semigroups III: modular equidistribution for numerical semigroups with arbitrarily many generators
Stephan Ramon Garcia, Mohamed Omar, Christopher O'Neill +1
For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrar…
Multidimensional Schinzel-type theorems for multiplicative functions near prime tuples
Stephan Ramon Garcia, Gabe Udell, Jiahui Yu
Assuming Dickson's conjecture, we obtain multidimensional analogues of recent results on the behavior of certain multiplicative arithmetic functions near twin-prime arguments. This…
The Prime Number Theorem as a Capstone in a Complex Analysis Course
Stephan Ramon Garcia
We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for…
Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials
Albrecht Boettcher, Stephan Ramon Garcia, Mohamed Omar +1
We employ the fact certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symm…
Factorization length distribution for affine semigroups II: asymptotic behavior for numerical semigroups with arbitrarily many generators
Stephan Ramon Garcia, Mohamed Omar, Christopher O'Neill +1
For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational r…
Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function
Stephan Ramon Garcia, Florian Luca, Kye Shi +1
Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies for a majority of twin-primes pairs and that the reverse inequality holds for…
Partially isometric matrices: a brief and selective survey
Stephan Ramon Garcia, Matthew Okubo Patterson, William T. Ross
We survey a variety of results about partially isometric matrices. We focus primarily on results that are distinctly finite-dimensional. For example, we cover a recent solution to…