arXiv subjects

Lucile Devin

Publications and source records attributed to Lucile Devin.

12 recordsLinked to original sources

arXiv2026

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$

We study the average analytic rank in the family of $L$-functions $L(s, E_d)$ associated with the elliptic curves $E_d : y^2=x^3-dx$, as $d$ varies over fourth-power-free odd integers. Since this is a family of curves with complex multiplication, we have $L(s, E_d)=L(s - \frac12, \xi_d)$, where $\xi_d$ is a Hecke character over $\mathbb{Z}[i]$. Assuming the Generalized Riemann Hypothesis, we compute the one-level density of the low-lying zeros of this family for test functions whose Fourier transform is supported in $(-\frac35, \frac35)$. As a consequence, we obtain the upper bound $\frac{13}{6}$ for the average analytic rank $r(E_d)$ over the family. Under the additional assumption of a conjecture on the distribution of quartic Gauss sums at prime elements (a quartic analogue of Patterson's conjecture for cubic Gauss sums), we extend the admissible support to $(-1, 1)$ and improve the upper bound for the average analytic rank to $\frac32$. Both results imply that a positive proportion of twists satisfy $r(E_d) =1$, while the second also yields a positive proportion of twists with $r(E_d)=0$.

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arXiv2025

Low-lying zeros in families of Maass form L-functions: an extended density theorem

We study the one-level density of low-lying zeros in the family of Maass form $L$-functions of prime level $N$ tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in $(-\frac32,\frac32)$. In this paper, we extend the unconditional admissible support to $(-\frac{15}8,\frac{15}8)$. The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet $L$-functions. Assuming the Grand Density Conjecture, we extend the admissible support to $(-2,2)$. In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of $L$-functions attached to holomorphic forms.

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arXiv2024

One-level densities in families of Gr\"ossencharakters associated to CM elliptic curves

We study the low-lying zeros of a family of $L$-functions attached to the CM elliptic curve $E_d \;:\; y^2 = x^3 - dx$, for each odd and square-free integer $d$. Specifically, upon writing the $L$-function of $E_d$ as $L(s-\frac12, \xi_d)$ for the appropriate Gr\"ossencharakter $\xi_d$ of conductor $\mathfrak{f}_d$, we consider the collection $\mathcal{F}_d$ of $L$-functions attached to $\xi_{d,k}$, $k \geq 1$, where for each integer $k$, $\xi_{d, k}$ denotes the primitive character inducing $\xi_d^k$. We observe that $25\%$ of the $L$-functions in $\mathcal{F}_d$ have negative root number. $\mathcal{F}_d$ is thus not one of the essentially homogeneous families of the Universality Conjecture of Sarnak, Shin and Templier, with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one-level density in the family of $L$-functions in $\mathcal{F}_{d}$ with conductor at most $K^2 \mathrm N (\mathfrak{f}_d)$, we find that $\mathcal{F}_d$ naturally decomposes into subfamilies: more specifically, a collection of symplectic ($L(s, \xi_{d,k})$ for $k \equiv \alpha \bmod 8$, $\alpha$ even) and orthogonal ($L(s, \xi_{d,k})$ for $k \equiv \alpha \bmod 8$, $\alpha$ odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of $\log (K^2 \mathrm N(\mathfrak{f}_d))$.

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arXiv2023

Exceptional biases in counting primes over functions fields

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

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arXiv2022

Extending the unconditional support in an Iwaniec-Luo-Sarnak family

We study the harmonically weighted one-level density of low-lying zeros of $L$-functions in the family of holomorpic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz--Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this admissible support to $(-\Theta_k,\Theta_k)$, where $\Theta_2 = 1.866\dots$ and $\Theta_k$ tends monotonically to $2$ as $k$ tends to infinity. This is asymptotically as good as the best known GRH result. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions.

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arXiv2021

Lemke Oliver and Soundararajan bias for consecutive sums of two squares

In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several results related to averages of the Hardy--Littlewood constant in the context of sums of two squares.

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arXiv2021

Discrepancies in the distribution of Gaussian primes

Motivated by questions of Fouvry and Rudnick on the distribution of Gaussian primes, we develop a very general setting in which one can study inequities in the distribution of analogues of primes through analytic properties of infinitely many $L$-functions. In particular, we give a heuristic argument for the following claim : for more than half of the prime numbers that can be written as a sum of two square, the odd square is the square of a positive integer congruent to $1 \bmod 4$.

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arXiv2019

Low-lying zeros in families of holomorphic cusp forms: the weight aspect

We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions $\phi$ satisfying the condition supp$(\widehat \phi) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat \phi$ reaches the point $1$. In particular the first of these terms involves the quantity $\widehat \phi(1)$ which appeared in previous work of Fouvry--Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of GRH for $\text{GL}(2)$ automorphic $L$-functions. Finally, when supp$(\widehat \phi)\subset (-1,1)$ we obtain an unconditional estimate which is significantly more precise than the prediction of the $L$-functions Ratios Conjecture.

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arXiv2019

Limiting properties of the distribution of primes in an arbitrarily large number of residue classes

We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\pi(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_0,a_1,\ldots,a_D$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of $x$ real for which the inequalities $\pi(x;q,a_0)>\pi(x;q,a_1)> \ldots >\pi(x;q,a_D)$ are simultaneously satisfied admits a logarithmic density.

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arXiv2018

Chebyshev's bias for products of irreducible polynomials

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon.

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arXiv2017

Chebyshev's bias for analytic L-functions

In this paper we discuss the generalizations of the concept of Chebyshev's bias from two perspectives. First we give a general framework for the study of prime number races and Chebyshev's bias attached to general $L$-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet $L$-functions and Hasse--Weil $L$-functions of elliptic curves over $\mathbf{Q}$. This also apply to new Chebyshev's bias phenomena that were beyond the reach of the previously known cases. In addition we weaken the required hypotheses such as GRH or linear independence properties of zeros of $L$-functions. In particular we establish the existence of the logarithmic density of the set $\lbrace x\geq 2 : \sum_{p\leq x} \lambda_{f}(p) \geq 0 \rbrace$ for coefficients $(\lambda_{f}(p))$ of general $L$-functions conditionally on a much weaker hypothesis than was previously known.

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arXiv2016

On the congruence class modulo prime numbers of the number of rational points of a variety

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in \mathbf{Z}$, we study the set $\bigcap_{i=1}^{n}\lbrace p\in\mathcal{P}-\Sigma_{X}, N_{X}(p)\neq a_{i}\ [\bmod\ p]\rbrace$ where $\Sigma_{X}$ is the finite set of primes of bad reduction for $X$. In case $\dim X\leq 3$, we show the set is either empty or has positive lower-density. We also address the question of the size of the smallest prime in that set. Using sieve methods, we obtain for example an upper bound for the size of the least prime of $\lbrace p\in\mathcal{P}, p\nmid N_{X}(p)\rbrace$ on average in particular families of hyperelliptic curves.

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