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Alessandro Fazzari

Publications and source records attributed to Alessandro Fazzari.

13 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

Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros

We consider the value distribution of the logarithm of the Riemann zeta function on the critical line, weighted by the local statistics of zeta zeros. We show that, with appropriate normalization, it satisfies a complex Central Limit Theorem, provided that the Fourier support of the test function in the linear statistics is sufficiently small. For the imaginary part, we extend this support condition up to its natural barrier under the Riemann Hypothesis. Finally, we prove that the correlation between $\log \zeta$ and the one-level density, while negligible on the level of Selberg's Central Limit Theorem, only decays at a rather slow rate if the Riemann Hypothesis is assumed. Our results can be viewed as a combination of Selberg's Central Limit Theorem with work of Hughes and Rudnick on mock-Gaussian behavior of the local statistics.

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arXiv2025

On products of sets of natural density one

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers $A$ and $B$ have natural density $1$, then their product set $A \cdot B := \{ab : a \in A, b \in B\}$ also has natural density $1$. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set $A$ of density 1 such that $A\cdot A$ has a ''large'' complement.

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arXiv2024

The third moment of the logarithm of zeta and a twisted pair correlation conjecture

We prove precise conditional estimates for the third moment of the logarithm of the Riemann zeta function, refining what is implied by the Selberg central limit theorem, both for the real and imaginary parts. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which explains the interaction of a prime power with Montgomery's pair correlation function. We believe this to be of independent interest, and devote substantial effort to its justification. Namely, we prove this conjecture on a certain range unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.

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arXiv2023

On the joint second moment of zeta and its logarithmic derivative

Assuming the Riemann Hypothesis, Goldston, Gonek and Montgomery \cite{GGM} studied the second moment of the log-derivative of $\zeta$, shifted away from the half line by $a/\log T$, and its connection with the pair correlation conjecture. In this paper, we consider a weighted version of this problem, where the average is tilted by $|\zeta(\frac{1}{2}+it)|^2$. More precisely, we provide an upper and a lower bound for the second moment of zeta times its logarithmic derivative, $a/\log T$ away from the critical line.

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arXiv2022

A weighted one-level density of the non-trivial zeros of the Riemann zeta-function

We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by $|\zeta(\frac12+it)|^{2k}$ for $k=1$ and, for test functions with Fourier support in $(-\frac12,\frac12)$, for $k=2$. As a consequence, for $k=1,2$, we deduce under the Riemann hypothesis that $T(\log T)^{1-k^2+o(1)}$ non-trivial zeros of $\zeta$, of imaginary parts up to $T$, are such that $\zeta$ attains a value of size $(\log T)^{k+o(1)}$ at a point which is within $O(1/\log T)$ from the zero.

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arXiv2022

Averages of long Dirichlet polynomials with modular coefficients

We study the moments of $L$-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a \textit{long} Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindel\"of Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

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arXiv2022

Hyperbolic angles from Heegner points

We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To prove this, we establish a connection between lattice points and algebraic integers in the associated field having norm of a special form and satisfying a congruence condition. As a by-product of this, we obtain an explicit formulation of the classical hyperbolic circle problem as a shifted convolution sum for the function that counts the number of algebraic integers with given norm. Along the way, we also prove a lower bound for shifted B-numbers, which is done by sieve methods.

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arXiv2021

Consecutive real quadratic fields with large class numbers

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as possible.

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arXiv2021

A weighted one-level density of families of $L$-functions

This paper is devoted to a weighted version of the one-level density of the non-trivial zeros of $L$-functions, tilted by a power of the $L$-function evaluated at the central point. Assuming the Riemann Hypothesis and the ratio conjecture, for some specific families of $L$-functions we prove that the same structure suggested by the density conjecture holds also in this weighted investigation, if the exponent of the weight is small enough. Moreover we speculate about the general case, conjecturing explicit formulae for the weighted kernels.

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arXiv2021

A weighted central limit theorem for $\log|ζ(1/2+it)|$

Under the Riemann Hypothesis, we show that as $t$ varies in $T\leq t \leq 2T$, the distribution of $\log|ζ(1/2+it)|$ with respect to the measure $|ζ(1/2+it)|^2dt$ is approximately normal with mean $\log\log T$ and variance $\frac{1}{2}\log\log T$.

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arXiv2021

Weighted value distributions of the Riemann zeta function on the critical line

We prove a central limit theorem for $\log|ζ(1/2+it)|$ with respect to the measure $|ζ^{(m)}(1/2+it)|^{2k}dt$ ($k,m\in\mathbb N$), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure $|ζ(1/2+it+iα)|^{2k}dt$, with $α\in(-1,1)$. Finally we prove unconditionally the analogue result in the random matrix theory context.

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