arXiv subjects

Youness Lamzouri

Publications and source records attributed to Youness Lamzouri.

At least 19 recordsLinked to original sources

arXiv2026

A new proof that more than $2/3$ of the zeros of the Riemann zeta function are simple and on the critical line

We obtain a new, conceptually simpler, unconditional proof that more than $67.25\%$ of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least $83.62\%$ of the non-trivial zeros are distinct. Our approach also yields two new unconditional estimates on simple zeros and zeros on the critical line. More precisely, we prove that the proportion of zeros that are simple or lie on the critical line (or both) is at least $88.76\%$, and that the average of the proportions of simple zeros and of zeros on the critical line is at least $83.62\%$. A proof of the bounds for simple zeros on the critical line and for distinct zeros was very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, whereas our two additional estimates are neither stated nor proved in the Claude paper. The argument produced by Claude is technically intricate, and its main mechanism is not immediately transparent. It combines several ingredients from linear algebra, including a finite-dimensional matrix representation of Weil's Hermitian form and a rank--trace inequality for Hermitian matrices, with a second moment calculation over the zeros using the explicit formula. Our new approach proceeds by replacing the entire finite-dimensional matrix framework by a single Hilbert space inequality, which allows for a direct application of Montgomery's theorem on the pair correlation of zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.

math.NT↗

arXiv2026

The distribution of the maximum of cubic character sums

For a primitive Dirichlet character $χ\pmod q$ we let \[M(χ):= \frac{1}{\sqrt{q}}\max_{1\leq t \leq q} \Big|\sum_{n \leq t} χ(n) \Big|.\] In this paper, we investigate the distribution of $M(χ)$, as $χ$ ranges over primitive cubic characters $χ\pmod q$ with $(q,3)=1$ and $q\leq Q$. Our first result gives an estimate for the proportion of such characters for which $M(χ)>V$, in a uniform range of $V$, which is best possible under the assumption of the Generalized Riemann Hypothesis. In particular, we show that the distribution of large cubic character sums behaves very differently from those in the family of non-principal characters modulo a large prime, and the family of quadratic characters. We also investigate the location of the number $N_χ$ where the maximum of $|\sum_{n\leq N} χ(n)|$ is attained, and show the surprising result that for almost all primitive cubic characters $χ\pmod q$ with $M(χ)>V$, $N_χ/q$ is very close to a reduced fraction with a large denominator of size $(\log V)^{1/2+o(1)}$. This contradicts the common belief that for an even character $χ$, $N_χ/q$ is located near a rational of small denominator and gives a striking difference with the case of even characters in the other two families mentioned above, for which $N_χ/q\approx 1/3$ or $2/3$ for almost all even $χ$. Furthermore, in the case of cubic characters, the works of Granville-Soundararajan, Goldmakher, and Lamzouri-Mangerel show that if $M(χ)$ is large, then $χ$ pretends to be $ξ(n)n^{it}$ for some small $t$, where $ξ$ is an odd character of small conductor $m$. We show that for almost all such characters, we have $M(χ)=m^{-1/2+o(1)}\big|L(1+it, χ\overlineξ)\big|.$

math.NT↗

arXiv2026

A note on large values of Dirichlet $L$-functions for characters of fixed order at $1/2<σ\leq 1$

In this note, we use a simple argument to show the existence of large values of conjecturally sharp size for Dirichlet $L$-functions attached to primitive characters of fixed order at $σ\in (1/2, 1]$. More precisely, for every fixed integer $g\geq 2$ we prove the existence of a primitive character $χ$ of order $g$ and conductor $Q\asymp x$ such that $|L(1,χ)| \geq e^γ\left(\log\log x+\log\log\log x-\log(2\log g)+o(1)\right). $ We also show that for every fixed $1/2<σ<1$ there exists a primitive character $χ$ of order $g$ and conductor $Q\asymp x$ such that $\log |L(σ,χ)| \geq \left(C_g(σ)+o(1)\right) (\log x)^{1-σ}(\log\log x)^{-σ}, $ for some explicit positive constant $C_g(σ).$ Previously, such bounds were known only conditionally on the Generalized Riemann Hypothesis, and even then only in the special cases $g=2$ and $g=3$.

math.NT↗

arXiv2026

An omega result for the least negative Hecke eigenvalue

We establish the existence of many holomorphic Hecke eigenforms $f$ of large weight $k$ for the full modular group, for which the least positive integer $n_f$ such that $λ_f(n_f)<0$ satisfies $n_f \ge (\log k)^{1-o(1)}.$ This is believed to be best possible up to the $o(1)$ term in the exponent, and improves on a result of Kowalski, Lau, Soundararajan and Wu, who showed that, when restricted to primes, the least prime $p$ such that $λ_f(p)<0$ can be as large as $(\log k)^{1/2+o(1)}$. We also discuss an extension of our result to primitive holomorphic cusp forms of weight $k$ and squarefree level $N\geq 1$.

math.NT↗

arXiv2026

Sharp omega results for the divisor and circle problems

We establish omega results for the divisor and circle problems that are conjecturally sharp, while also determining the sign of the large values obtained. This improves on the work of Soundararajan and on the subsequent independent refinements of Sourmelidis and Mahatab, and gives the first improvement on Hafner's 1981 $Ω_+$ result for the divisor problem and his $Ω_-$ result for the circle problem. The main new ingredient is a resonance method which works directly with the phase appearing in the Voronoï summation formula. This is achieved by replacing the usual positive kernels by a one-sided sectorial kernel, namely the density of a Gamma distribution, whose Fourier transform lies in a suitable sector of the complex plane.

math.NT↗

arXiv2026

Large values of exponential sums with multiplicative coefficients

In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form $\sum_{n\leq x}f(n)e(nα)$ where $f$ is a $1$-bounded multiplicative function and $α\in\mathbb R$, close to the conjectured $\ll \frac{x}{\sqrt{q}}+ \frac{x}{\log x}$ where $α$ is best approximated by $|α-a/q|\leq 1/(qx)$, showing their results to be ``best-possible'' by observing that the first part of their bound is more-or-less attained when $f(n)=χ(n), α=\frac aq$ where $χ$ is a primitive character mod $q$, and the second part when $f(p)=e(-αp)$ for all large primes $p$. La Bretèche and Granville proved that when $α$ lies on a major arc the exponential sum is significantly smaller unless $f$ ``pretends to be'' $χ(n)n^{it}$ for some character $χ$ and real number $|t|<\log x$; and herein we prove that when $α$ lies on a minor arc, the exponential sum is significantly smaller unless $f(p)$ pretends to be $e(-hpα)$ for primes $p\leq x$ for some bounded integer $h$. We also study exponential sums $\sum_{n\leq x, P^+(n)\leq y} f(n) e(nα)$ restricted to $y$-smooth (or $y$-friable) integers $n$. We conjecture that this sum is $\ll \frac{Ψ(x, y)}{\sqrt{q}}+ \frac{\sqrt{xy}}{\log x} $ in a wide range of parameters, show that if true this is best possible, and prove an upper bound in a wide range that is only slightly weaker than the conjecture. Finally we study the logarithmically weighted exponential sums $\sum_{n\leq x} \frac{f(n)}{n} e(nα)$. We conjecture that this sum is $\ll \frac{\log x}{\sqrt{q}}+\log q$ in a wide range of parameters, show that if true this is best possible, and prove an upper bound in a wide range that is only slightly weaker than the conjecture. Along the way, we will prove various technical results about multiplicative functions which may be of use elsewhere.

math.NT↗

arXiv2026

Real zeros of $L'(s, χ_d)$

In 1990, Baker and Montgomery conjectured that $L'(s,χ_d)$ has $\asymp \log\log |d|$ real zeros in the interval $[1/2,1]$ for almost all fundamental discriminants $d$. The study of these zeros was motivated by their connection to real zeros of Fekete polynomials and to sign changes of the character sums $\sum_{n\leq x}χ_d(n)$. Recent work of Klurman, Lamzouri, and Munsch shows that the number of such zeros is $\gg (\log\log |d|)/(\log\log\log\log |d|)$ for almost all $d$, thereby establishing the conjectured lower bound up to the factor $\log\log\log\log |d|$. In this paper, we prove that for almost all fundamental discriminants $d$, $L'(s,χ_d)$ has at most $(\log\log |d|)(\log\log\log |d|)$ real zeros in $[1/2,1]$, thus resolving the Baker-Montgomery conjecture up to a factor of $\log\log\log |d|$. We also give a quantitative upper bound on the exceptional set of discriminants. Furthermore, we show, conditionally on certain natural assumptions, that $100\%$ of these zeros lie away from $1/2$.

math.NT↗

arXiv2025

Erdős's integer dilation approximation problem and GCD graphs

Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{α\in\mathcal{A}\cap[1,x]}\frac{1}α>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many pairs $(α, β)\in \mathcal{A}^2$ such that $α\neq β$ and $|nα-β| <\varepsilon$ for some positive integer $n$. This resolves a problem of Erdős from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation.

math.NT↗

arXiv2024

On the distribution of the error terms in the divisor and circle problems

We study the distribution functions of several classical error terms in analytic number theory, focusing on the remainder term in the Dirichlet divisor problem $Δ(x)$. We first bound the discrepancy between the distribution function of $Δ(x)$ and that of a corresponding probabilistic random model, improving results of Heath-Brown and Lau. We then determine the shape of its large deviations in a certain uniform range, which we believe to be the limit of our method, given our current knowledge about the linear relations among the $\sqrt{n}$ for square-free positive integers $n$. Finally, we obtain similar results for the error terms in the Gauss circle problem and in the second moment of the Riemann zeta function on the critical line.

math.NT↗

arXiv2024

Sign changes of short character sums and real zeros of Fekete polynomials

We discuss a general approach producing quantitative bounds on the number of sign changes of the weighted sums $$\sum_{n\le x}f(n)w_n$$ where $f:\mathbb{N}\to \mathbb{R}$ is a family of multiplicative functions and $w_n\in\mathbb{R}$ are certain weights.As a consequence, we show that for a typical fundamental discriminant $D,$ the partial sums of the real character $χ_D$ change sign $\gg (\log\log D)/\log\log\log \log D$ times on very short initial interval (which goes beyond the range in Vinogradov's conjecture). We also prove that the number of real zeros (localized away from $1$) of the Fekete polynomial associated to a typical fundamental discriminant $D$ is $\gg \frac{\log\log D}{\log\log\log\log D}.$ This comes close to establishing a conjecture of Baker and Montgomery which predicts $\asymp \log \log D$ real zeros. Finally, the same approach shows that almost surely for large $x\ge 1$, the partial sums $\sum_{n\le y}f(n)$ of a (Rademacher) random multiplicative function exhibit $\gg \log \log x/\log \log\log\log x$ sign changes on the interval $[1,x].$ These results rely crucially on uniform quantitative estimates for the joint distribution of $-\frac{L'}{L}(s, χ_D)$ at several points $s$ in the vicinity of the central point $s=1/2$, as well as concentration results for $\log L(s,χ_D)$ in the same range, which we establish. In the second part of the paper, we obtain, for large families of discriminants, ``non-trivial" upper bounds on the number of real zeros of Fekete polynomials, breaking the square root bound. Finally, we construct families of discriminants with associated Fekete polynomials having no zeros away from $1.$

math.NT↗

arXiv2024

The limiting distribution of Legendre paths

Let $p$ be a prime number and $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol modulo $p$. The \emph{Legendre path} attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}} \sum_{n\leq j} \left(\frac{n}{p}\right)$ for $0\leq j\leq p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $Q\leq p\leq 2Q$, when $Q$ is large. Our main result shows that as $Q \to \infty$, these paths converge in law, in the space of real-valued continuous functions on $[0, 1]$, to a certain random Fourier series constructed using Rademacher random completely multiplicative functions. This was previously proved by the first author under the assumption of the Generalized Riemann Hypothesis.

math.NT↗

arXiv2024

An effective Linear Independence conjecture for the zeros of the Riemann zeta function and applications

Using the same heuristic argument leading to the Lang-Waldschmidt Conjecture in the theory of linear forms in logarithms, we formulate an effective version of the Linear Independence conjecture for the ordinates of the non-trivial zeros of the Riemann zeta function. Then assuming this conjecture, we obtain Omega results for the error term in the prime number theorem, which are conjectured to be best possible by Montgomery. This was claimed to appear in Monach's thesis by Montgomery and Vaughan in their classical book, but such a result or its proof is nowhere to be found in this thesis or in the litterature. Moreover, if in addition to this effective Linear Independence conjecture we assume a conjecture of Gonek and Hejhal on the negative moments of the derivative of the Riemann zeta function, we can show that the summatory function of the Möbius function $M(x)$ is $Ω_{\pm}\left(\sqrt{x}(\log\log\log x)^{5/4}\right)$. This conditionally resolves a part of a conjecture of Gonek.

math.NT↗

arXiv2024

The distribution of large quadratic character sums and applications

In this paper, we investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants $|d|\leq x$. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to $x$, and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to $3$ modulo $4$. Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of non-principal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols.

math.NT↗

arXiv2023

$L_q$ norms and Mahler measure of Fekete polynomials

We show that the distribution of the values of Fekete polynomials $F_p$ on the unit circle is governed, as $p\to\infty$, by an explicit limiting (non-Gaussian) random point process.This allows us to prove that the Mahler measure of $F_p$ satisfies $$M_0(F_p)\sim k_0\sqrt{p},$$ as $p\to\infty$ where $k_0=0.74083\dots,$ thus solving an old open problem. Further, we obtain an asymptotic formula for all moments $\|F_p\|_q$ with $0<q<\infty,$ resolving another open problem and improving previous results of Günther and Schmidt (who treated the case $q=2k,$ $k\in\mathbb{N}$).

math.NT↗

arXiv2023

The number of zeros of linear combinations of $L$-functions near the critical line

In this paper, we investigate the zeros near the critical line of linear combinations of $L$-functions belonging to a large class, which conjecturally contains all $L$-functions arising from automorphic representations on $\text{GL}(n)$. More precisely, if $L_1, \dots, L_J$ are distinct primitive $L$-functions with $J\ge 2$, and $b_j$ are any nonzero real numbers, we prove that the number of zeros of $F(s)=\sum_{j\leq J} b_j L_j(s)$ in the region $\text{Re}(s)\geq 1/2+1/G(T)$ and $\text{Im}(s)\in [T, 2T]$ is asymptotic to $K_0 T G(T)/\sqrt{\log G(T)}$ uniformly in the range $ \log \log T \leq G(T)\leq (\log T)^ν$, where $K_0$ is a certain positive constant that depends on $J$ and the $L_j$'s. This establishes a generalization of a conjecture of Hejhal in this range. Moreover, the exponent $ν$ verifies $ν\asymp 1/J$ as $J$ grows.

math.NT↗

arXiv2021

The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$-adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$-periodic complex valued functions which satisfy our conditions, but for which the Pólya-Vinogradov inequality is sharp.

math.NT↗

arXiv2021

Large deviations of sums of random variables

In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some of the results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences $p^{it}$, $χ(p)$, $χ_d(p)$, $λ_f(p)$, and $\text{Kl}_q(a-n, b)$; where $p$ ranges over the primes, $t$ varies in a large interval, $χ$ varies among all characters modulo $q$, $χ_d$ varies over quadratic characters attached to fundamental discriminants $|d|\leq x$, $λ_f(n)$ are the Fourier coefficients of holomorphic cusp forms $f$ of (a large) weight $k$ for the full modular group, and $\text{Kl}_q(a, b)$ are the normalized Kloosterman sums modulo a large prime $q$, where $a, b$ vary in $(\mathbb{F}_q)^{\times}$.

math.NT↗

arXiv2021

Low pseudomoments of Euler products

In this paper, we determine the order of magnitude of the $2q$-th pseudomoment of powers of the Riemann zeta function $ζ(s)^α$ for $0<q\le 1/2$ and $0< α<1$, completing the results of Bondarenko, Heap and Seip, and of Gerspach. Our results also apply to more general Euler products satisfying certain conditions.

math.NT↗