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Javier Pliego

Publications and source records attributed to Javier Pliego.

10 recordsLinked to original sources
arXiv2025

Local divisor correlations in almost all short intervals

Let $ k,l \geq 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum $\sum_{x 0$ be a small fixed number and let $Φ(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{Φ(x)}$ and $(\log x)^{1000k\log k}\leq H_2\leq H_1^{1-\varepsilon }$, the expected asymptotic formula holds for almost all $x\in[X,2X]$ and almost all $1\leq h\leq H_2$.

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arXiv2025

On Vu's theorem in Waring's problem for thinner sequences

Let $k\in \mathbb{N}$ and $s\geq k(\log k+3.20032)$. Let $\mathbb{N}_{0}^{k}$ be the set of $k$-th powers of nonnegative integers. Assume that $ψ$ is an increasing function tending to infinity with $ψ(x)=o(\log x)$ and satifying some regularity conditions. Then, there exists a subsequence $\mathfrak{X}_{k}=\mathfrak{X}_{k}(s)\subset\mathbb{N}_{0}^{k}$ for which the number of representations $R_{s}(n;\mathfrak{X}_{k})$ of each $n\in\mathbb{N}$ as $$n=x_{1}^{k}+\ldots+x_{s}^{k}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_{i}^{k}\in\mathfrak{X}_{k}$$ satisfies the asymptotic formula $$ R_{s}(n;\mathfrak{X}_{k})\sim \mathfrak{S}(n)ψ(n)$$ for almost all natural numbers $n$, with $\mathfrak{S}(n)$ being the singular series associated to Waring's problem. If moreover $s\geq k(\log k+4.20032)$ the above conclusion holds for almost all $n\in [X,X+\log X]$ as $X\to\infty$. Let $T(k)$ be the least natural number for which it is known that all large integers are the sum of $T(k)$ $k$-th powers of natural numbers. We also show for $k\geq 14$ and every $s\geq T(k)$ the existence of a sequence $\mathfrak{X}_{k}'\subset \mathbb{N}_{0}^{k}$ satisfying $$R_{s}(n;\mathfrak{X}_{k}')\asymp \log n$$ for every sufficiently large $n$. The latter conclusion sharpens a result of Wooley and addresses a question of Vu.

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arXiv2024

On the Erdős-Turán Conjecture and the growth of $B_{2}[g]$ sequences

When $g\in\mathbb{N}$ we say that $A\subset\mathbb{N}$ is a $B_{2}[g]$ sequence if every $m\in\mathbb{N}$ has at most $g$ distinct representations of the shape $m=b_{1}+b_{2}$ with $b_{1}\leq b_{2}$ and $b_{1},b_{2}\in A$. We show for every $0<\varepsilon<1$ that whenever $g>\frac{1}{\varepsilon}$ then there is a $B_{2}[g]$ sequence $A$ having the property that every sufficiently large $n\in\mathbb{N}$ can be written as $$n=a_{1}+a_{2}+a_{3},\ \ \ \ \ \ \ \ \ a_{3}\leq n^{\varepsilon}\ \ \ \ \ \ \ \ \ a_{i}\in A,$$ and satisfying for large $x$ the estimate $$\lvert A\cap [1,x]\rvert\gg x^{g/(2g+1)}.$$ The above lower bound improves upon earlier results of Cilleruelo and of Erdős and Renyi.

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arXiv2022

Twisted mixed moments of the Riemann zeta function

We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape $P(\log T) T^{C}$ for a suitable constant $C<1$ and a polynomial $P(x)$. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$-conjecture.

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arXiv2022

Mixed moments of the Riemann zeta function

We analyse a collection of mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$ conjecture.

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arXiv2020

On Waring's problem in sums of three cubes

We investigate the asymptotic formula for the number of representations of a large positive integer as a sum of $k$-th powers of integers represented as the sums of three positive cubes, counted with multiplicities. We also obtain a lower bound for the number of representations when the sums of three cubes are counted without multiplicities.

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arXiv2020

On Waring's problem in sums of three cubes for smaller powers

We give an upper bound for the minimum $s$ with the property that every sufficiently large integer can be represented as the sum of $s$ positive $k$-th powers of integers represented as the sum of three positive cubes for the cases $2\leq k\leq 4.$

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arXiv2020

Uniform bounds in Waring's problem over some diagonal forms

We investigate the existence of representations of every large positive integer as a sum of $k$-th powers of integers represented as certain diagonal forms. In particular, we consider a family of diagonal forms and discuss the problem of giving a uniform upper bound over the family for the number of variables needed to have such representations.

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arXiv2020

On squares of sums of three cubes

We show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.

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