arXiv subjects

Sebastian Zuniga Alterman

Publications and source records attributed to Sebastian Zuniga Alterman.

10 recordsLinked to original sources

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

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

Weighted sieves with switching

Weighted sieves are used to detect numbers with at most $S$ prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes $p$ such that $p+2$ has at most two prime factors), one can utilize the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes. As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

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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

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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arXiv2022

On the Atkinson formula for the $ζ$ function

Thanks to Littlewood (1922) and Ingham (1928), we know the first two terms of the asymptotic formula for the square mean integral value of the Riemann zeta function $ζ$ on the critical line. Later, Atkinson (1939) presented this formula with an error term of order $O(\sqrt{T}\log^{2}(T))$, which we call the Atkinson formula. Following the latter approach and the work of Titchmarsh (1986), we present an explicit version of the Atkinson formula, improving on a recent bound by Simonič (2020). Moreover, we extend the Atkinson formula to the range $\Re(s)\in\left[\frac{1}{4},\frac{3}{4}\right]$, giving an explicit bound for the square mean integral value of $ζ$ and improving on a bound by Helfgott and the authors (2019). We use mostly classical tools, such as the approximate functional equation and the explicit convexity bounds of the zeta function given by Backlund (1918).

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arXiv2022

Dynamics of $\mathscr{B}$-free systems generated by Behrend sets. I

We study the complexity of $\mathscr{B}$-free subshifts which are proximal and of zero entropy. Such subshifts are generated by Behrend sets. The complexity is shown to achieve any subexponential growth and is estimated for some classical subshifts (prime and semiprime subshifts). We also show that $\mathscr{B}$-admissible subshifts are transitive only for coprime sets $\mathscr{B}$ which allows one to characterize dynamically the subshifts generated by the Erdös sets.

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arXiv2022

Explicit $L^2$ bounds for the Riemann $ζ$ function

Explicit bounds on the tails of the zeta function $ζ$ are needed for applications, notably for integrals involving $ζ$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $ζ$. Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large $T$, is based on classical lines, starting with an approximation to $ζ$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<σ\leq 1$ and are strong enough to be of practical use for the rigorous computation of improper integrals. We also present bounds for the $L^{2}$ norm of $ζ$ in $[1,T]$ for $0\leqσ\leq 1$.

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arXiv2021

On a logarithmic sum related to a natural quadratic sieve

We study the sum $Σ_q(U)=\sum_{\substack{d,e\leq U\\(de,q)=1}}\frac{μ(d)μ(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, so that a continuous, monotonic and explicit version of Selberg's sieve can be stated. Thanks to Barban-Vehov (1968), Motohashi (1974) and Graham (1978), it has been long known, but never explicitly, that $Σ_1(U)$ is asymptotic to $\log(U)$. In this article, we discover not only that $Σ_q(U)\sim\frac{q}{φ(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also we find a closed-form expression for its secondary order term of $Σ_q(U)$, a constant $\mathfrak{s}_q$, which we are able to estimate explicitly when $q=v\in\{1,2\}$. We thus have $Σ_v(U)= \frac{v}{φ(v)}\log(U)-\mathfrak{s}_v+O_v^*\left(\frac{K_v}{\log(U)}\right)$, for some explicit constant $K_v > 0$, where $\mathfrak{s}_1=0.60731\ldots$ and $\mathfrak{s}_2=1.4728\ldots$. As an application, we show how our result gives an explicit version of the Brun-Titchmarsh theorem within a range.

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arXiv2020

Explicit averages of square-free supported functions: to the edge of the convolution method

We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and observe how the nature of this method gives error term estimations of order $X^{-δ}$, where $δ$ belongs to an open real positive set $I$. In order to have a better error estimation, a natural question is whether or not we can achieve an error term of critical order $X^{-δ_0}$, where $δ_0$, the critical exponent, is the right hand endpoint of $I$. We reply positively to that question by presenting a new method that improves qualitatively almost all instances of the convolution method under some regularity conditions; now, the asymptotic estimation of averages of well-behaved square-free supported arithmetic functions can be given with its critical exponent and a reasonable explicit error constant. We illustrate this new method by analyzing a particular average related to the work of Ramaré--Akhilesh (2017), which leads to notable improvements when imposing non-trivial coprimality conditions.

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