Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2408.04143

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

Daniel R. Johnston·Nicol Leong·Sebastian Tudzi
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{Ω(n)}, \end{equation*} where $Ω(n)$ 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 $|W(x)| 0$.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel R. Johnston, Nicol Leong, Sebastian Tudzi. 2024-09-08. New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$. https://arxiv.org/abs/2408.04143

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On single-variable Witten zeta functions of rank two and three

By introducing a novel integration kernel for the Mellin transform, we uncover many previously unknown and intriguing properties of the Witten zeta functions of rank two and three. Detailed results concerning their pole locations, residues, and special values are obtained. We propose a non-trivial conjecture regarding their derivatives at the origin, which seems to encode deep information about the root system. We also discuss their behavior at negative integers, highlighting a connection with Eisenstein series and a $p$-adic observation.

math.NT↗
arXiv2026

On graded Lie algebras associated to once-punctured elliptic curves with complex multiplication

We study a graded Lie algebra arising from the Galois action on the pro-$p$ fundamental group of a once-punctured elliptic curve with complex multiplication. Among other things, we provide a minimal generating set of the rationalized Lie algebra under suitable assumptions. The proof is based on a slight variant of the theory of weighted completion of profinite groups developed by Hain and Matsumoto.

math.NT↗
arXiv2026

Burgess-type volume dependent bounds for character sums over $\mathbb{F}_{p^n}$

We establish a Burgess-type bound for short multiplicative character sums over finite fields $\mathbb{F}_{p^n}$. Let \[ B=\left\{\sum_{i=1}^{n}x_iω_i: N_i+1\le x_i\le N_i+H_i,1\le i\le n\right\}\subseteq\mathbb{F}_{p^n}, \] where $1\le H_i\le p$ for all $1\le i\le n$, and the side lengths satisfy $H_1\le H_2\le\cdots\le H_n.$ We prove that if the side lengths satisfy certain lower bounds in terms of the two largest side lengths, then a nontrivial cancellation occurs in the character sum over the boxes. This generalizes the work of Gabdullin \cite{GB} in dimensions $n=2,3$ to arbitrary dimension. This also generalizes the character sum estimate of Konyagin \cite{Kon} where each of the side lengths of the boxes are greater than $p^{1/4}$. The proof combines techniques from the geometry of numbers, multiplicative energy estimates, and Katz's bounds for multiplicative character sums.

math.NT↗