O. Ramar'e

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20242026

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math.NT2024

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

Olivier RamarÃ©

Let the $A$ -cusps of a dense subset $P^{*} \in [N, N]$ of primes be points $I^\pm \in R / Z$ that are such that $|\sum_{\substack{p\in\mathcal{P}^*}} e(Î±p)|…

math.NT2024

A spectral resolution of the large sieve

Olivier RamarÃ©

The quadratic form $V (I ¨ , Q) = \sum_{q \sim Q} \sum_{a mod^{*} q} ∣ S (I ¨ , a / q) ∣^{2}$ and its eigenvalues are well understood when $Q = o (N)$ , while $V (I ¨ , Q)$ is expected to behave like a…

math.NT2024

From explicit estimates for the primes to explicit estimates for the MÃ¶bius function II

Olivier RamarÃ©, Sebastian Zuniga Alterman

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}Î¼(n)\B…

math.NT2024

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),…

math.NT2024

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

Let $I^»$ the Barban--Vehov weights, defined in $(1)$ . Let $X \geq z_{1} \geq 100$ and $z_{2} = z_{1}^{I ¨} $ for some $I ¨ > 1$ . We prove that \begin{equation*} \sum_{n\le X}\frac{1}{n}\Bigl(\sum_{\su…

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

A spectral resolution of the large sieve

From explicit estimates for the primes to explicit estimates for the MÃ¶bius function II

Convolution of periodic multiplicative functions and the divisor problem

An L2-bound for the Barban-Vehov weights

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

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