arXiv subjects

Enrique Treviño

Publications and source records attributed to Enrique Treviño.

7 recordsLinked to original sources

arXiv2026

Supercharacter theory and applications to Ramanujan sums over a finite Frobenius ring

The theory of Ramanujan sums has been playing a fundamental role in several subfields of mathematics. They appear in the theory of special values of zeta functions, spectral graph theory, representation theory, and analytic number theory. Recent work has shown that classical Ramanujan sums can also be interpreted as a super-Fourier transform via the theory of supercharacters for the ring $\mathbb{Z}/n$. In this article, building upon our recent work on supercharacters over an arbitrary finite Frobenius ring, we explore additional arithmetical properties of Ramanujan sums. Our approach provides a unified framework that generalizes various results in the literature regarding these sums. As a by-product, we also describe a new criterion for determining when a finite commutative ring is Frobenius, which could be of independent interest to the algebra community.

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arXiv2026

Explicit Burgess inequalities for cubefree moduli

Burgess proved that for $χ_q$ a primitive Dirichlet character modulo $q$ with $q$ cubefree, $\sum_{M< n\le M+N}χ_q(n)= O\left(N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+ε}\right)$ for all integers $r\ge1.$ More recently, explicit versions with prime moduli $q$ were computed by Booker, McGown, Treviño, and Francis, with applications to finding the least $k$-th power residue, and bounding the size of Dirichlet $L$-functions just to name a few. Jain-Sharma, Khale, and Liu proved an explicit estimate for $r=2.$ We improve their explicit constant for $r = 2$ and compute an explicit Burgess bound for cubefree $q$ for $r\ge 3$.

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arXiv2025

Polynomial densities and Heilbronn's criterion

Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn's criterion applies to a polynomial $f$ if it applies to the number field $K=\mathbb{Q}[x]/(f)$ generated by $f$. Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd(p-1,n)=1$. Let ${F}_{p,n}$ denote the collection of monic polynomials $f\in\mathbb{Z}[x]$ of degree $n$ that are Eisenstein at the prime $p$. We order our polynomials by the natural height $\mathrm{Ht}(f)$. Define $δ_{p,n}(X)$ to be the proportion of polynomials $f\in {F}_{p,n}$ with $\mathrm{Ht}(f)\leq X$ for which Heilbronn's criterion applies. One has $$\liminf_{X\to\infty}δ_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\varepsilon(p)\right\}\,,$$ where $\varepsilon(p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to\infty$ uniformly in $n$. We also give a version of this result where we weaken the condition on $\gcd(p-1,n)$. As a corollary, we show that given an integer $n\geq 2$, a positive proportion of Eisenstein polynomials of degree $n$ fail to generate norm-Euclidean fields.

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arXiv2021

Generalizing parking functions with randomness

Consider $n$ cars $C_1, C_2, \ldots, C_n$ that want to park in a parking lot with parking spaces $1,2,\ldots,n$ that appear in order. Each car $C_i$ has a parking preference $α_i \in \{1,2,\ldots,n\}$. The cars appear in order, if their preferred parking spot is not taken, they take it, if the parking spot is taken, they move forward until they find an empty spot. If they do not find an empty spot, they do not park. An $n$-tuple $(α_1, α_2, \ldots, α_n)$ is said to be a parking function, if this list of preferences allows every car to park under this algorithm. For an integer $k$, we say that an $n$-tuple is a $k$-Naples parking function if the cars can park with the modified algorithm, where park $C_i$ backs up $k$-spaces (one by one) if their spot is taken before trying to find a parking spot in front of them. We introduce randomness to this problem in two ways: 1) For the original parking function definition, for each car $C_i$ that has their preference taken, we decide with probability $p$ whether $C_i$ moves forwards or backwards when their preferred spot is taken; 2) For the $k$-Naples definition, for each car $C_i$ that has their preference taken, we decide with probability $p$ whether $C_i$ backs up $k$ spaces or not before moving forward. In each of these models, for an $n$-tuple $α\in\{1,2,\ldots,n\}^n$, there is now a probability that the model ends in all cars parking or not. For each of these random models, we find a formula for the expected value. Furthermore, for the second random model, in the case $k =1$, $p=1/2$, we prove that for any $1\le t\le 2^{n-2}$, there is exactly one $α\in\{1,2,\ldots,n\}^n$ such that the probability that $α$ parks is $(2t-1)/2^{n-1}$.

math.CO↗

arXiv2020

On the sum of $k$-th powers in terms of earlier sums

For $k$ a positive integer let $S_k(n) = 1^k + 2^k + \cdots + n^k$, i.e., $S_k(n)$ is the sum of the first $k$-th powers. Faulhaber conjectured (later proved by Jacobi) that for $k$ odd, $S_k(n)$ could be written as a polynomial of $S_1(n)$; for example $S_3(n) = S_1(n)^2$. We extend this result and prove that for any $k$ there is a polynomial $g_k(x,y)$ such that $S_k(n) = g(S_1(n), S_2(n))$. The proof yields a recursive formula to evaluate $S_k(n)$ as a polynomial of $n$ that has roughly half the number of terms as the classical one.

math.NT↗

arXiv2015

Resolving Grosswald's conjecture on GRH

In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.

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arXiv2015

The Burgess inequality and the least k-th power non-residue

The Burgess inequality is the best upper bound we have for the character sum $S_χ(M,N) = \sum_{M<n\le M+N} χ(n).$ Until recently, no explicit estimates had been given for the inequality. In 2006, Booker gave an explicit estimate for quadratic characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than $10^{140}$. Both of their explicit estimates are on restricted ranges. In this paper we prove an explicit estimate that works for any $M$ and $N$. We also improve McGown's estimates in a slightly narrower range, getting explicit estimates for characters of any order. We apply the estimates to the question of how large must a prime $p$ be to ensure that there is a $k$-th power non-residue less than $p^{1/6}$.

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