S. Garcia

activity

20092026

most citedMathematical and physical aspects of complex symmetric operators

149 citations · 319 across the 56 of their papers we have counts for

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26 papers · 1 filter

math.NT2022

The error term in the truncated Perron formula for the logarithm of an $L$ -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 $L$ -function. All our constants are explicit.

math.NT2021

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 $p - 1 = d k$ and fixed $k$ , we compute the fourth absolute moments for all but finitely many primes $p$ . For $d$ …

math.NT2021

Explicit estimates for Artin $L$ -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 $L$ -functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerica…

math.NT2020

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…

math.NT2020

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 $f \in Z [x]$ . We use an explicit version o…

math.NT2020★ 9 cited

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 $x \geq 2$ . Our error terms are explicitly bounde…

math.NT2020

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.

math.NT2020

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…

math.NT2019

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 $ϕ (p - 1) \geq ϕ (p + 1)$ for a majority of twin-primes pairs $p, p + 2$ and that the reverse inequality holds for…

math.NT2018

$p$ -adic quotient sets II: quadratic forms

Christopher Donnay, Stephan Ramon Garcia, Jeremy Rouse

For $A \subseteq {1, 2, \dots}$ , we consider $R (A) = {a / a^{'} : a, a^{'} \in A}$ . If $A$ is the set of nonzero values assumed by a quadratic form, when is $R (A)$ dense in the $p$ -adic…

math.NT2018

The Bateman-Horn Conjecture: Heuristics, History, and Applications

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, Stephan Ramon Garcia

The Bateman-Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green-T…

math.NT2018

Supercharacters, elliptic curves, and the sixth moment of Kloosterman sums

Stephan Ramon Garcia, George Todd

We connect the sixth power moment of Kloosterman sums to elliptic curves. This yields an elementary proof that $K_{u}$ with $p ∤ u$ are $O (p^{2/3})$ .

math.NT2017

Primitive root biases for prime pairs I: existence and non-totality of biases

Stephan Ramon Garcia, Florian Luca, Timothy Schaaff

We study the difference between the number of primitive roots modulo $p$ and modulo $p + k$ for prime pairs $p, p + k$ . Assuming the Bateman-Horn conjecture, we prove the existence of s…

math.NT2017

On the difference in values of the Euler totient function near prime arguments

Stephan Ramon Garcia, Florian Luca

We prove unconditionally that for each $ℓ \geq 1$ , the difference $ϕ (p - ℓ) - ϕ (p + ℓ)$ is positive for $50%$ of odd primes $p$ and negative for $50%$ .

math.NT2016

$p$ -adic quotient sets

Stephan Ramon Garcia, Yu Xuan Hong, Florian Luca +4

For $A \subseteq N$ , the question of when $R (A) = {a / a^{'} : a, a^{'} \in A}$ is dense in the positive real numbers $R_{+}$ has been examined by many authors over the y…

math.NT2015★ 1 cited

Spherical 2-designs and lattices from Abelian groups

Albrecht Boettcher, Simon Eisenbarth, Lenny Fukshansky +2

We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spheric…

math.NT2015★ 1 cited

Visual properties of generalized Kloosterman sums

Paula Burkhardt, Alice Zhuo-Yu Chan, Gabriel Currier +3

For a positive integer $m$ and a subgroup $Λ$ of the unit group $(Z / m Z)^{\times}$ , the corresponding generalized Kloosterman sum is the function $K(a,b,m,Λ) = \sum_…

math.NT2015

Lattices from Hermitian function fields

Albrecht Boettcher, Lenny Fukshansky, Stephan Ramon Garcia +1

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are genera…

math.NT2015

Gauss' hidden menagerie: from cyclotomy to supercharacters

Stephan Ramon Garcia, Trevor Hyde, Bob Lutz

Gaussian periods, when viewed appropriately, exhibit a dazzling and eclectic host of visual qualities. This brief survey reviews the historical context and summarizes our current k…

math.NT2014★ 2 cited

On lattices generated by finite Abelian groups

Albrecht Boettcher, Lenny Fukshansky, Stephan Ramon Garcia +1

This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields.…

math.NT2013

Four quotient set gems

Bryan Brown, Michael Dairyko, Stephan Ramon Garcia +2

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotie…

math.NT2013

A supercharacter approach to Heilbronn sums

Stephan Ramon Garcia, Bob Lutz

Various algebraic properties of Heilbronn's exponential sum can be deduced through the use of supercharacter theory, a novel extension of classical character theory due to Diaconis…

math.NT2013

The graphic nature of the symmetric group

J. L. Brumbaugh, Madeleine Bulkow, Luis Alberto Garcia +3

We investigate a remarkable class of exponential sums which are derived from the symmetric groups and which display a diverse array of visually appealing features. Our interest in…

math.NT2012

The graphic nature of Gaussian periods

William Duke, Stephan Ramon Garcia, Bob Lutz

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework within which to study certain exponential sums of interest in number theory.…

The error term in the truncated Perron formula for the logarithm of an L-function

Moments of Gaussian Periods and Modified Fermat Curves

Explicit estimates for Artin L-functions: Duke's short-sum theorem and Dedekind Zeta Residues

A Gallery of Gaussian Periods

An effective analytic formula for the number of distinct irreducible factors of a polynomial

Unconditional Explicit Mertens' Theorems for Number Fields and Dedekind Zeta Residue Bounds

Explicit Mertens' Theorems for Number Fields

Multidimensional Schinzel-type theorems for multiplicative functions near prime tuples

Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function

p-adic quotient sets II: quadratic forms

The Bateman-Horn Conjecture: Heuristics, History, and Applications

Supercharacters, elliptic curves, and the sixth moment of Kloosterman sums

Primitive root biases for prime pairs I: existence and non-totality of biases

On the difference in values of the Euler totient function near prime arguments

p-adic quotient sets

Spherical 2-designs and lattices from Abelian groups

Visual properties of generalized Kloosterman sums

Lattices from Hermitian function fields

Gauss' hidden menagerie: from cyclotomy to supercharacters

On lattices generated by finite Abelian groups

Four quotient set gems

A supercharacter approach to Heilbronn sums

The graphic nature of the symmetric group

The graphic nature of Gaussian periods

The error term in the truncated Perron formula for the logarithm of an $L$ -function

Explicit estimates for Artin $L$ -functions: Duke's short-sum theorem and Dedekind Zeta Residues

$p$ -adic quotient sets II: quadratic forms

$p$ -adic quotient sets