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of 56
pro vyhledávání: '"Ramare, Olivier"'
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}\mu(n)\Bigg| &\le \frac{0.006688\,X}{\log X},&&\text{for } X\ge 1\,798\,118,
Externí odkaz:
http://arxiv.org/abs/2408.05969
Let $\lambda$ the Barban--Vehov weights, defined in $(1)$. Let $X\ge z_1\ge100$ and $z_2=z_1^\tau$ for some $\tau>1$. We prove that \begin{equation*} \sum_{n\le X}\frac{1}{n}\Bigl(\sum_{\substack{d|n}}\lambda_d\Bigr)^2 \le f(\tau)\frac{\log X}{\log (
Externí odkaz:
http://arxiv.org/abs/2405.12662
We study the sums $\sum_{n\le X, (n,q)=1}\frac{\mu(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$ and give asymptotic estimations in an explicit manner. In order to do so, we produce a large family of arithm
Externí odkaz:
http://arxiv.org/abs/2312.05138
Autor:
Ramaré, Olivier
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 $\chi$ is the non-princi
Externí odkaz:
http://arxiv.org/abs/2308.10876
Autor:
Ramaré, Olivier
We provide numerical bounds for $\Sigma(X)=\sum_{\substack{d_1,d_2\le X}}\frac{\mu(d_1)\mu(d_2)}{[d_1,d_2]}$. We show in particular that $0\le \Sigma(X)\le 17/25$ for every $X\ge2$.
Externí odkaz:
http://arxiv.org/abs/2308.07632
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
Externí odkaz:
http://arxiv.org/abs/2305.06260
Autor:
Ramaré, Olivier
The quadratic form $V(\varphi,Q)=\sum_{q\sim Q}\sum_{a\mod^* q}|S(\varphi,a/q)|^2$ and its eigenvalues are well understood when $Q=o(\sqrt{N})$, while $V(\varphi,Q)$ is expected to behave like a Riemann sum when $N=o(Q)$. The behavior in the range $Q
Externí odkaz:
http://arxiv.org/abs/2303.04409
Autor:
Ramaré, Olivier
This note proposes a probabilistic language-free proof of the famous Croot-Laba-Sisask Lemma. In between, we do the same for the Khintchine and Marcinkiewicz-Zygmund inequalities and explicitate the implied constants.
Externí odkaz:
http://arxiv.org/abs/2212.05920
Given a primitive, non-CM, holomorphic cusp form $f$ with normalized Fourier coefficients $a(n)$ and given an interval $I\subset [-2, 2]$, we study the least prime $p$ such that $a(p)\in I$ . This can be viewed as a modular form analogue of Vinogrado
Externí odkaz:
http://arxiv.org/abs/2208.14786
Publikováno v:
Journal of Number Theory, Volume 243, February 2023, Pages 13-37
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 narro
Externí odkaz:
http://arxiv.org/abs/2208.06602