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Julio Andrade

Publications and source records attributed to Julio Andrade.

13 recordsLinked to original sources
arXiv2026

Correlations of zeros of a family of $L$-functions in function fields with symplectic symmetry

In this paper, we adapt the framework developed by Mason and Snaith to investigate the $n$-level density of zeros in the context of function fields. Specifically, we derive explicit formulas for the $n$-level density of zeros in families of quadratic Dirichlet $L$-functions associated with hyperelliptic curves of genus $g$ over the finite field $\mathbb{F}_{q}$. Employing Mason and Snaith's method, we obtain precise expressions for the $1$-level density in these families and extend the approach to higher-level densities. Furthermore, we apply the method to derive formulas for the $n$-level density of zeros in families of $L$-functions associated with prime characters. Our results are consistent with the findings of Andrade, Jung, and Shamesaldeen in the case $n=1$.

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arXiv2020

The Moments and statistical Distribution of Class number of Primes over Function Fields

We investigate the moment and the distribution of $L(1,\x_P),$ where $\x_P$ varies over quadratic characters associated to irreducible polynomials $P$ of degree $2g+1$ over $\mathbb{F}_q[T]$ as $g\to\infty$. In the first part of the paper we compute the integral moments of the class number $h_{P}$ associated to quadratic function fields with prime discriminants $P$ and this is done by adapting to the function field setting some of the previous results carried out by Nagoshi in the number field setting. In the second part of the paper we compute the complex moments of of $L(1,\x_P)$ in large uniform range and investigate the statistical distribution of the class numbers by introducing a certain random Euler product. The second part of the paper is based on recent results carried out by Lumley when dealing with square-free polynomials.

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arXiv2019

Mean values of derivatives of $L$-functions in function fields: IV

In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $\mu\geq1$ an integer, we compute the mean value of the $\mu$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.

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arXiv2019

On Additive Divisor Sums and minorants of divisor functions

We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are fixed. Our results relate the parameter $A$ to the lengths of arithmetic progressions in which $d_k(n)$ is uniformly distributed. As applications to additive divisor sums, we establish new lower bounds and a new equivalent condition for the conjectured asymptotic. We also prove a Tauberian theorem for general additive divisor sums.

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arXiv2018

The Integral Moments and Ratios of Quadratic Dirichlet $L$-Functions over Monic Irreducible Polynomials in $\mathbb{F}_{q}[T]$

In this paper we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of $L$-functions. We also adapt to the function setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of $L$-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet $L$-functions $L(s,\chi_{P})$ where the character $\chi$ is defined by the Legendre symbol for polynomials in $\mathbb{F}_{q}[T]$ with $\mathbb{F}_{q}$ a finite field of odd cardinality and the averages are taken over all monic and irreducible polynomials $P$ of a given odd degree. As an application we also compute the formula for the one-level density for the zeros of these $L$-functions.

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arXiv2018

Small zeros of Dirichlet $L$-functions of quadratic characters of prime modulus

In this paper, we investigate the distribution of the imaginary parts of zeros near the real axis of Dirichlet $L$-functions associated to the quadratic characters $\chi_{p}(\cdot)=(\cdot |p)$ with $p$ a prime number. Assuming the Generalized Riemann Hypothesis (GRH), we compute the one-level density for the zeros of this family of $L$-functions under the condition that the Fourier transform of the test function is supported on a closed subinterval of $(-1,1)$. We also write down the ratios conjecture for this family of $L$-functions a la Conrey, Farmer and Zirnbauer and derive a conjecture for the one-level density which is consistent with the main theorem of this paper and with the Katz-Sarnak prediction and includes lower order terms. Following the methods of \"Ozl\"uk and Snyder, we prove that GRH implies $L(\frac{1}{2},\chi_p)\neq 0$ for at least $75\%$ of the primes.

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arXiv2015

Rudnick and Soundararajan's Theorem for Function Fields

In this paper we prove a function field version of a theorem by Rudnick and Soundararajan about lower bounds for moments of quadratic Dirichlet $L$-functions. We establish lower bounds for the moments of quadratic Dirichlet $L$--functions associated to hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_{q}$ in the large genus $g$ limit.

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arXiv2015

A Simple Proof of the Mean Value of $\left|K_{2}(\mathcal{O})\right|$ in Function Fields

Let $F$ be a finite field of odd cardinality $q$, $A=F[T]$ the polynomial ring over $F$, $k=F(T)$ the rational function field over $F$ and $\mathcal{H}$ the set of square-free monic polynomials in $A$ of degree odd. If $D\in\mathcal{H}$, we denote by $\mathcal{O}_{D}$ the integral closure of $A$ in $k(\sqrt{D})$. In this note we give a simple proof for the average value of the size of the groups $K_{2}(\mathcal{O}_{D})$ as $D$ varies over the ensemble $\mathcal{H}$ and $q$ is kept fixed. The proof is based on character sums estimates and in the use of the Riemann hypothesis for curves over finite fields.

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arXiv2013

Newman's conjecture in various settings

De Bruijn and Newman introduced a deformation of the Riemann zeta function $ζ(s)$, and found a real constant $Λ$ which encodes the movement of the zeros of $ζ(s)$ under the deformation. The Riemann hypothesis (RH) is equivalent to $Λ\le 0$. Newman made the conjecture that $Λ\ge 0$ along with the remark that "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Newman's conjecture is still unsolved, and previous work could only handle the Riemann zeta function and quadratic Dirichlet $L$-functions, obtaining lower bounds very close to zero (for example, for $ζ(s)$ the bound is at least $-1.14541 \cdot 10^{-11}$, and for quadratic Dirichlet $L$-functions it is at least $-1.17 \cdot 10^{-7}$). We generalize the techniques to apply to automorphic $L$-functions as well as function field $L$-functions. We further determine the limit of these techniques by studying linear combinations of $L$-functions, proving that these methods are insufficient. We explicitly determine the Newman constants in various function field settings, which has strong implications for Newman's quantitative version of RH. In particular, let $\mathcal D \in \bbZ[T]$ be a square-free polynomial of degree 3. Let $D_p$ be the polynomial in $\bbF_p[T]$ obtained by reducing $\mathcal D$ modulo $p$. Then the Newman constant $Λ_{D_p}$ equals $\log \frac{|a_p(\mathcal D)|}{2\sqrt{p}}$; by Sato--Tate (if the curve is non-CM) there exists a sequence of primes such that $\lim_{n \to\infty} Λ_{D_{p_n}} = 0$. We end by discussing connections with random matrix theory.

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arXiv2013

Special Sets of Primes in Function Fields

When investigating the distribution of the Euler totient function, one encounters sets of primes P where if p is in P then r is in P for all r|(p-1). While it is easy to construct finite sets of such primes, the only infinite set known is the set of all primes. We translate this problem into the function field setting and construct an infinite such set in F_p[x] whenever p is equivalent to 2 or 5 modulo 9.

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arXiv2013

Hilbert-Pólya Conjecture, Zeta-Functions and Bosonic Quantum Field Theories

The original Hilbert and Pólya conjecture is the assertion that the non-trivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However the suggestion of Hilbert and Pólya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by Number Theory. In this paper we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularization, is that the various Dirichlet series attached to such sequences - such as those which are sums over "primes" of $(\mathrm{norm} \ P)^{-s}$ have a natural boundary, i.e., they cannot be continued beyond the line $\mathrm{Re}(s)=0$. The main argument is that once the regularized determinant of a Laplacian is meromorphic in $s$, it follows that the series considered above cannot be a regularized determinant. In other words we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed.

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arXiv2012

A Note on the Mean Value of $L$--functions in Function Fields

An asymptotic formula for the sum $\sum L(1,χ)$ is established for a family of hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$ as $g\rightarrow\infty$ making use of the analogue of the approximate functional equation for such $L$--functions. As a corollary, we obtain a formula for the average of the class number of the associated rings $\mathbb{F}_{q}[T,sqrt{D}]$.

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