arXiv subjects

Sebastian Tudzi

Publications and source records attributed to Sebastian Tudzi.

4 recordsLinked to original sources

arXiv2026

On the Summatory Function of $d_3(n)$

In this article, we refine the method of our earlier work with N. Paloj{\"a}rvi to obtain a sharper explicit bound for the error term $\Delta_{3}(x)$ associated with the summatory function of $d_{3}(n)$. We prove that \begin{equation*} |\Delta_3(x)| < \begin{cases} 0.6901\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 3.682\cdot 10^{31}\le x < 4.133\cdot 10^{87},\\[4pt] 0.2067\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 4.133\cdot 10^{87} \le x < 1.597\cdot 10^{98},\\[4pt] 0.1947\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & x \ge 1.597\cdot 10^{98}. \end{cases} \end{equation*} These explicit results improve the exponent of $x$ from $2/3$, due to Tudzi, and $859/1400$, due to Paloj{\"a}rvi and Tudzi, to $1/2$, giving the best known bound for all $x\ge 3.682\cdot 10^{31}$.

math.NT↗

arXiv2026

Explicit and Effective Estimates for the error term in the Generalised Divisor Problem

In this article, we obtain effective estimates for the error term $\Delta_{k}(x)$ for all integers $k \geq2$, and completely explicit estimates for integers $k \in [3,9]$. The explicit results improve the powers of $x$ appearing in the known explicit bounds for $\Delta_{k}(x)$, and the effective bounds provide a method to derive such bounds for all integers $k\geq 3$.

math.NT↗

arXiv2025

On the Generalised Divisor Problem

In this paper, we apply the Dirichlet convolution method to \begin{equation*} T_{k}(x)=\sum_{n \leq x} d_{k}(n), \end{equation*} for $k\ge 3$, where $d_{k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer factors. We prove that for $k=3$, the error term $|\Delta_3(x)|< 2.968x^{2/3}\log^{1/3}x$ for all $x\ge 2$. This improves the best-known explicit result established by Bordell{\`e}s for all $x\ge 2$. We extend this for all $k>3$ and obtain an explicit error term of the form $\Delta_{k}(x)=O\left(x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right)$.

math.NT↗

arXiv2024

New bounds and progress towards a conjecture on the summatory function of $(-2)^{\Omega(n)}$

In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{\Omega(n)}, \end{equation*} where $\Omega(n)$ counts the number of prime factors of $n$, with multiplicity. We prove $W(x)=O(x)$, and in particular, that $|W(x)|<2260x$ for all $x\geq 1$. This result provides new progress towards a conjecture of Sun, which asks whether $|W(x)| 0$.

math.NT↗