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Youness Lamzouri

Publications and source records attributed to Youness Lamzouri.

51 records · Page 3Linked to original sources
arXiv2014

Discrepancy bounds for the distribution of the Riemann zeta-function and applications

We investigate the distribution of the Riemann zeta-function on the line $\Re(s)=σ$. For $\tfrac 12 < σ\le 1$ we obtain an upper bound on the discrepancy between the distribution of $ζ(s)$ and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of $ζ(s)$ inside of the critical strip, strengthening a previous result of the first author. As an application of these results we obtain the first effective error term for the number of solutions to $ζ(s) = a$ in a strip $\tfrac12 < σ_1 < σ_2 < 1$. Previously in the strip $\tfrac 12 < σ< 1$ only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg) and to the right of the abscissa of absolute convergence (due to Matsumoto). In general our results are an improvement of the classical Bohr-Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function.

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arXiv2012

Large even order character sums

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis.

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arXiv2011

Lower bounds on odd order character sums

A classical result of Paley shows that there are infinitely many quadratic characters $χ\mod{q}$ whose character sums get as large as $\sqrt{q}\log \log q$; this implies that a conditional upper bound of Montgomery and Vaughan cannot be improved. In this paper, we derive analogous lower bounds on character sums for characters of odd order, which are best possible in view of the corresponding conditional upper bounds recently obtained by the first author.

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arXiv2011

Large deviations of the limiting distribution in the Shanks-Rényi prime number race

Let $q\geq 3$, $2\leq r\leq ϕ(q)$ and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis, M. Rubinstein and P. Sarnak showed that the vector-valued function $E_{q;a_1,...,a_r}(x)=(E(x;q,a_1),..., E(x;q,a_r)),$ where $E(x;q,a)= \frac{\log x}{\sqrt{x}}(ϕ(q)π(x;q,a)-π(x))$, has a limiting distribution $μ_{q;a_1,...,a_r}$ which is absolutely continuous on $\mathbb{R}^r$. Under the same assumptions, we determine the asymptotic behavior of the large deviations $μ_{q;a_1,...,a_r}(||\vx||>V)$ for different ranges of $V$, uniformly as $q\to\infty.$

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arXiv2011

The Shanks-Rényi prime number race with many contestants

Under certain plausible assumptions, M. Rubinstein and P. Sarnak solved the Shanks--Rényi race problem, by showing that the set of real numbers $x\geq 2$ such that $π(x;q,a_1)>π(x;q,a_2)>...>π(x;q,a_r)$ has a positive logarithmic density $δ_{q;a_1,...,a_r}$. Furthermore, they established that if $r$ is fixed, $δ_{q;a_1,...,a_r}\to 1/r!$ as $q\to \infty$. In this paper, we investigate the size of these densities when the number of contestants $r$ tends to infinity with $q$. In particular, we deduce a strong form of a recent conjecture of A. Feuerverger and G. Martin which states that $δ_{q;a_1,...,a_r}=o(1)$ in this case. Among our results, we prove that $δ_{q;a_1,...,a_r}\sim 1/r!$ in the region $r=o(\sqrt{\log q})$ as $q\to\infty$. We also bound the order of magnitude of these densities beyond this range of $r$. For example, we show that when $\log q\leq r\leq ϕ(q)$, $δ_{q;a_1,...,a_r}\ll_ε q^{-1+ε}$.

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arXiv2011

The distribution of short character sums

Let $χ$ be a non-real Dirichlet character modulo a prime $q$. In this paper we prove that the distribution of the short character sum $S_{χ,H}(x)=\sum_{x< n\leq x+H} χ(n)$, as $x$ runs over the positive integers below $q$, converges to a two-dimensional Gaussian distribution on the complex plane, provided that $\log H=o(\log q)$ and $H\to\infty$ as $q\to\infty$. Furthermore, we use a method of Selberg to give an upper bound on the rate of convergence.

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arXiv2011

Randomness of character sums modulo $m$

Using a probabilistic model, based on random walks on the additive group $\mathbb{Z}/m\mathbb{Z}$, we prove that the values of certain real character sums are uniformly distributed in residue classes modulo $m$.

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arXiv2011

On the distribution of extreme values of zeta and $L$-functions in the strip $1/2<σ<1$

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=σ$ where $1/2<σ<1$. We consider the Riemann zeta function $ζ(s)$ in the $t$-aspect, Dirichlet $L$-functions in the $q$-aspect, and $L$-functions attached to primitive holomorphic cusp forms of weight $2$ in the level aspect. For each family we show that the $L$-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We also prove new $Ω$-results for quadratic Dirichlet $L$-functions (predicted to be best possible by the probabilistic model) conditionally on GRH, and other results related to large moments of $ζ(σ+it)$.

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arXiv2011

Prime number races with three or more competitors

Fix an integer $r\geq 3$. Let $q$ be a large positive integer and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. In this paper, we establish an asymptotic formula for the logarithmic density $δ_{q;a_1,...,a_r}$ of the set of real numbers $x$ such that $π(x;q,a_1)>π(x;q,a_2)>...>π(x;q,a_r),$ as $q\to\infty$; conditionally on the assumption of the Generalized Riemann Hypothesis GRH and the Grand Simplicity Hypothesis GSH. Several applications concerning these prime number races are then deduced. Indeed, comparing with a recent work of D. Fiorilli and G. Martin for the case $r=2$, we show that these densities behave differently when $r\geq 3$. Another consequence of our results is the fact that, unlike two-way races, biases do appear in races involving three of more squares (or non-squares) to large moduli. Furthermore, we establish a conjecture of M. Rubinstein and P. Sarnak (on biased races) in certain cases where the $a_i$ are assumed to be fixed and $q$ is large. We also prove that a conjecture of A. Feuerverger and G. Martin concerning "bias factors" (which follows from the work of Rubinstein and Sarnak for $r=2$) does not hold when $r\geq 3$. Finally, we use a variant of our method to derive Fiorilli and Martin asymptotic formula for the densities in two-way races.

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arXiv2010

Extreme values of $\arg L(1,χ)$

In this paper we study the distribution of extreme values of $\arg L(1,χ)$, as $χ$ varies over primitive characters modulo a large prime $q$.

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arXiv2010

Distribution of values of $L$-functions at the edge of the critical strip

We prove several results on the distribution of values of $L$-functions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power $L$-functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these $L$-functions) at $s=1$, functions in the Selberg class (in the height aspect), and quadratic twists of a fixed $GL(m)/{\Bbb Q}$-automorphic cusp form at $s=1$.

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arXiv2008

The two dimensional distribution of values of $ζ(1+it)$

We prove several results on the distribution function of $ζ(1+it)$ in the complex plane, that is the joint distribution function of $\argζ(1+it)$ and $|ζ(1+it)|$. Similar results are also given for $L(1,χ)$ (as $χ$ varies over non-principal characters modulo a large prime $q$).

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arXiv2008

On the number of linear forms in logarithms

Let $n$ be a positive integer. In this paper we estimate the size of the set of linear forms $b_1\log a_1 + b_2\log a_2+...+b_n\log a_n$, where $|b_i|\leq B_i$ and $1\leq a_i\leq A_i$ are integers, as $A_i,B_i\to \infty$.

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arXiv2005

Smooth values of the iterates of the Euler's Phi function

Let $ϕ(n)$ be the Euler-phi function, define $ϕ_0(n) = n$ and $ϕ_{k+1}(n)=ϕ(ϕ_{k}(n))$ for all $k\geq 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which $ϕ_k(n)$ is $y$-smooth, conditionally on a weak form of the Elliott-Halberstam conjecture.

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