Searcharxiv⌕ Search

arXiv subjects

Stephan Ramon Garcia

Publications and source records attributed to Stephan Ramon Garcia.

At least 55 records · Page 3Linked to original sources
arXiv2020

The Prime Number Theorem as a Capstone in a Complex Analysis Course

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 any instructor who seeks to use the prime number theorem for a series of capstone lectures, a scaffold for a series of guided exercises, or as a framework for an inquiry-based course. We require almost no knowledge of number theory, for our aim is to make a complete proof of the prime number theorem widely accessible to complex analysis instructors. In particular, we highlight the potential pitfalls and subtleties that may catch the instructor unawares when using more terse sources.

math.HO↗
arXiv2020

Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials

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 symmetric polynomials of even degree $2p$, the positivity of which is a classical result by D. B. Hunter. We extend Hunter's result to complete homogeneous symmetric polynomials of fractional degree, which are defined via Jacobi's bialternant formula. We show in particular that these polynomials have positive real part for real degrees $μ$ with $|μ-2p|< 1/2$. We also prove a positivity criterion for linear combinations of the classical complete homogeneous symmetric polynomials and a sufficient criterion for the positivity of linear combinations of products of such polynomials.

math.CO↗
arXiv2019

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

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 a small positive proportion of the twin primes. That is, $p$ tends to have more primitive roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{ϕ(p+1)}{ϕ(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{ϕ(p+1)}{ϕ(p)}$ and $\frac{σ(p+1)}{σ(p)}$, in which $σ$ denotes the sum-of-divisors function.

math.NT↗
arXiv2019

$p$-adic quotient sets II: quadratic forms

For $A \subseteq \{1,2,\ldots\}$, 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 numbers? We show that for a binary quadratic form $Q$, $R(A)$ is dense in $\mathbb{Q}_{p}$ if and only if the discriminant of $Q$ is a nonzero square in $\mathbb{Q}_{p}$, and for a quadratic form in at least three variables, $R(A)$ is always dense in $\mathbb{Q}_{p}$. This answers a question posed by several authors in 2017.

math.NT↗
arXiv2019

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

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-Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman-Horn conjecture, its applications, and its origins.

math.NT↗
arXiv2019

Partially isometric matrices: a brief and selective survey

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 the similarity problem for partial isometries. We also discuss the unitary similarity problem and several other results.

math.FA↗
arXiv2018

Factorization length distribution for affine semigroups I: numerical semigroups with three generators

Most factorization invariants in the literature extract extremal factorization behavior, such as the maximum and minimum factorization lengths. Invariants of intermediate size, such as the mean, median, and mode factorization lengths are more subtle. We use techniques from analysis and probability to describe the asymptotic behavior of these invariants. Surprisingly, the asymptotic median factorization length is described by a number that is usually irrational.

math.AC↗
arXiv2018

$p$-adic quotient sets

For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a' : a, a' \in A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by many authors over the years. In contrast, the $p$-adic setting is largely unexplored. We investigate conditions under which $R(A)$ is dense in the $p$-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed in this endeavor. We also pose many open questions that should be of general interest.

math.NT↗
arXiv2017

A supercharacter approach to Heilbronn sums

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-Isaacs and Andre. This perspective yields a variety of formulas and provides a method for computing the number of solutions to Fermat-type congruences.

math.NT↗
arXiv2017

Permutation invariant lattices

We say that a Euclidean lattice in $\mathbb R^n$ is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group $S_n$, i.e., if the lattice is closed under the action of some non-identity elements of $S_n$. Given a fixed element $τ\in S_n$, we study properties of the set of all lattices closed under the action of $τ$: we call such lattices $τ$-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio, which we studied in a recent paper. Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all $τ$-invariant lattices in $\mathbb R^n$ has positive co-dimension (and hence comprises zero proportion) for all $τ$ different from an $n$-cycle.

math.CO↗
arXiv2017

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

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 strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs $p,p+k$, the dominant inequality is reversed.

math.NT↗
arXiv2017

Primitive root bias for twin primes

Numerical evidence suggests that for only about $2\%$ of pairs $p,p+2$ of twin primes, $p+2$ has more primitive roots than does $p$. If this occurs, we say that $p$ is exceptional (there are only two exceptional pairs with $5 \leq p \leq 10{,}000$). Assuming the Bateman-Horn conjecture, we prove that at least $0.47\%$ of twin prime pairs are exceptional and at least $65.13\%$ are not exceptional. We also conjecture a precise formula for the proportion of exceptional twin primes.

math.NT↗
arXiv2017

Spherical 2-designs and lattices from Abelian groups

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 spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the structure of the automorphism groups of these lattices, strengthening our previous results in this direction.

math.NT↗