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Olivier Ramaré

Publications and source records attributed to Olivier Ramaré.

At least 19 recordsLinked to original sources
arXiv2026

The large sieve through Parseval, Large sifted sets are regular

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, we prove that sifted sets are either small or the attached Fourier polynomial is very regular at the origin. We also discuss the optimality of this approach and shall show in passing that $N\int_{-δ}^δ|S(β)|^2dβ\ge(1-2/\sqrt{δN})|S(0)|^2$ for any trigonometric polynomial of length~$N$.

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arXiv2026

The weighted large sieve through Parseval

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, improve on the weighted large sieve inequality. We compute in passing the $L^2$-norm of the Beurling-Selberg majorant of the characteristic function of an interval.

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arXiv2026

On a Möbius double sum

We study the double sum $S_\varepsilon(X)$$=$$\sum_{\substack{d,e\le X}}\frac{μ(d)μ(e)}{[d,e]^{1+\varepsilon}}$, which converges even in the case $\varepsilon=0$, where $μ$ denotes the Möbius function and $[d,e]$ is the least common multiple of $d$ and $e$. Such expressions arise naturally in analytic number theory, notably as the diagonal contribution in certain squared mean values, and they play a significant role in zero-density estimates for the Riemann zeta function and related $L$-functions. We establish uniform upper bounds for $S_\varepsilon(X)$ across various ranges of $X$, with particular emphasis on the case $\varepsilon$ close to $0^+$.

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arXiv2026

The weighted large sieve through Parseval

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, improve on the weighted large sieve inequality beyond what was heuristically accessible by the earlier approach. We also discuss the optimality of this approach.

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arXiv2026

Bounding the exponential sum on squares of some sifted sequences

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(α; N)=\sum_{n\le N}b(n)e(n^2α)$ satisfies \begin{equation*} \frac{S(α;N)}{N/\sqrt{\log N}} \ll N^ε(q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|α- a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences.

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arXiv2026

Möbius function and primes: an identity factory with applications

We investigate the sums $\sum_{n\le X, (n,q)=1}\frac{μ(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality $\sum_{n\le X}Λ(n)/n\le \log X$, valid for any $X\geq 1$.

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arXiv2025

Rudin Inequality, Chang Theorem, primes and squares

We prove that the set of large values of the trigonometric polynomial over a subset of density of the primes has some additive structure, similarly to what happens for subsets of densities in $\mathbb{Z}/{N}\mathbb{Z}$ but in a weaker form. To do so, we prove large sieve inequalities for \emph{dissociate sets} $\mathcal{X}$ of circle points and functions $f$ whose support~$S$ is finite and respectively in an interval, in the set of primes or in the set of squares. Set $T(f,x)=\sum_{n}f(n)\exp(2iπnx)$. These inequalities are of the shape $\sum_{x\in\mathcal{X}}|T(f,x)|^2\ll |S|\|f\|_2^2\log(8R/|S|)$ where $R$ is respectively $N$, $N/\log N$ and $\sqrt{N}$. The implied constants depend on the spacement between sumsets of~$\mathcal{X}$.

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arXiv2024

On the behaviour of $γ\Log p$ modulo 1

We prove non-trivial lower bounds for sums of type $\sum_{p\sim P}g(γ\Log p)$, where $g$ is a non-negative $2π$-periodical function and $γ$ is a given parameter. As an application we prove that $ζ(1+it)^{\pm1}\ll\Log\Log (9+|t|)$ and extend the zero-free region of the Riemann zeta-function.

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arXiv2024

Cusps of primes in dense subsequences -- Bypassing the $W$-trick

Let the $A$-cusps of a dense subset $\mathcal{P}^*\in[\sqrt{N},N]$ of primes be points $α\in\mathbb{R}/\mathbb{Z}$ that are such that $|\sum_{\substack{p\in\mathcal{P}^*}} e(αp)|\ge |\mathcal{P}^*|/A$. We establish that any $(1/N)$-well spaced subset of $A$-cusps contains at most $20A^2K\log(2A)$ points, where $K=N/(|\mathcal{P}^*|\log N)$. We further show that any $B$-cusps~$ξ$ is accompanied, when $B\le \sqrt{A}$, by a large proportion of $A$-cusps of the shape $ξ+(a/q)$. We conclude this study by showing that, given $A\ge2$, the characteristic function $1_{\mathcal{P}^*}$ may be decomposed in the form $1_{\mathcal{P}^*}=(V(z_0)\log N)^{-1}f^\flat +f^\sharp$ where the trigonometric polynomial of $f^\sharp$ takes only values $\le |\mathcal{P}^*|/A$, and~$f^\flat$ is a bounded non-negative function supported on the integers prime to $M$; the parameters $z_0$ and $M$ are given in terms of~$A$, while $V(z_0)=\prod_{p<z_0}(1-1/p)$. The function $f^\flat$ satisfies more regularity properties. In particular, its density with respect to the integers $\le N$ and coprime to~$M$ is again~$K$. This transfers questions on~$\mathcal{P}^*$ to problems on integers coprime to the modulus~$M$.

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arXiv2024

A spectral resolution of the large sieve

The quadratic form $V(φ,Q)=\sum_{q\sim Q}\sum_{a\mod^* q}|S(φ,a/q)|^2$ and its eigenvalues are well understood when $Q=o(\sqrt{N})$, while $V(φ,Q)$ is expected to behave like a Riemann sum when $N=o(Q)$. The behavior in the range $Q\in[\sqrt{N},100 N]$ is still mysterious. In the present work we present a full spectral analysis when $Q\ge N^{7/8}$ in terms of the eigenvalues of a one-parameter family of nuclear difference operators. We show in particular that (a smoothed version of) the quadratic form $V(φ,Q)$ may stay \emph{away} from $(6/π^2)Q\sum_n|φ_n|^2$ when $Q\asymp N$, though only on a vector space of positive but small dimension. An improved version of this paper, with the same title, will appear (2024 or 2025) in the Bulletin of the French Mathematical Society.

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arXiv2024

From explicit estimates for the primes to explicit estimates for the Möbius function II

We improve on all the results of [13] by incorporating the finite range computations performed since then by several authors. Thus we have \begin{align*} \Bigg|\sum_{n\le X}μ(n)\Bigg| &\le \frac{0.006688\,X}{\log X},&&\text{for } X\ge 1\,798\,118, \\\Bigg|\sum_{n\le X}\frac{μ(n)}{n}\Bigg| & \le \frac{0.010032}{\log X},&& \text{for } X\ge 617\,990. \end{align*} We also improve on the method described in [13] by a simple remark.

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arXiv2024

Convolution of periodic multiplicative functions and the divisor problem

We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and $1$-pretentious. We prove that if $f_1$ and $f_2$ belong to this class, then $\sum_{n\leq x}(f_1\ast f_2)(n)=Ω(x^{1/4})$. This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between $Δ(x)$ and $Δ(θx)$, where $θ$ is a fixed real number. We prove that there is a non-trivial correlation when $θ$ is rational, and a decorrelation when $θ$ is irrational. Moreover, if $θ$ has a finite irrationality measure, then we can make it quantitative this decorrelation in terms of this measure.

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arXiv2024

An $L^2$-bound for the Barban-Vehov weights

Let $λ$ the Barban--Vehov weights, defined in $(1)$. Let $X\ge z_1\ge100$ and $z_2=z_1^τ$ for some $τ>1$. We prove that \begin{equation*} \sum_{n\le X}\frac{1}{n}\Bigl(\sum_{\substack{d|n}}λ_d\Bigr)^2 \le f(τ)\frac{\log X}{\log (z_2/z_1)}, \end{equation*} for a completely determined function $f:(1,\infty)\to\mathbb{R}_{>0}$. In particular, we may take $f(2)=30$, saving more than a factor of $5$ on what was the best known result for $τ=2$. Two related estimates are also provided for general $τ>1$.

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arXiv2023

Explicit count of integral ideals of an imaginary quadratic field

We provide explicit bounds for the number of integral ideals of norms at most $X$ is $\mathbb{Q}[\sqrt{d}]$ when $d <0$ is a fundamendal discriminant with an error term of size $O(X^{1/3})$. In particular, we prove that, when $χ$ is the non-principal character modulo $3$ and $X\ge1$, we have $\sum_{n\le X}(1\starχ)(n) = \fracπ{3\sqrt{3}}X +O^*( 1.94\,X^{1/3})$, and that, when $χ$ is the non-principal character modulo $4$ and $X\ge1$, we have $\sum_{n\le X}(1\starχ)(n) = \frac{π}{4}X+ O^*( 1.4\,X^{1/3})$.

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arXiv2023

A Moebius sum

We provide numerical bounds for $Σ(X)=\sum_{\substack{d_1,d_2\le X}}\frac{μ(d_1)μ(d_2)}{[d_1,d_2]}$. We show in particular that $0\le Σ(X)\le 17/25$ for every $X\ge2$.

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arXiv2022

Counting ideals in ray classes

Let $\mathbf{K}$ be a number field and $\mathfrak{q}$ an integral ideal in $\mathcal{O}_{\mathbf{K}}$. A result of Tatuzawa from 1973, computes the asymptotic (with an error term) for the number of ideals with norm at most $x$ in a class of the narrow ray class group of $\mathbf{K}$ modulo $\mathfrak{q}$. This result bounds the error term with a constant whose dependence on $\mathfrak{q}$ is explicit but dependence on $\mathbf{K}$ is not explicit. The aim of this paper is to prove this asymptotic with a fully explicit bound for the error term.

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arXiv2022

Notes on restriction theory in the primes

TO BE PUBLISHED BY ISRAEL JOURNAL OF MATHEMATICS. We study the mean $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p e(xp)\bigr|^{\ell}$ when $\ell$ covers the full range $[2,\infty)$ and $\mathcal{X}\subset\mathbb{R}/\mathbb{Z}$ is a {well-spaced} set, providing a smooth transition from the case $\ell=2$ to the case $\ell>2$ and improving on the results of J.~Bourgain and of B.~Green and T.~Tao. A uniform Hardy-Littlewood property for the set of primes is established as well as a sharp upper bound for $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p e(xp)\bigr|^{\ell}$ when $\mathcal{X}$ is small. These results are extended to primes in \emph{any} interval in a last section, provided the primes are numerous enough therein.

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