Erdős Problems

Browse all problems

↑ ↓ to move↵ to openesc to close

Loading problem…

Status

On this page

Status

Topics

Terms Privacy info@plasma.ai

Problem 444 — Erdős Problems

Problem 444

Let $A\subseteq\mathbb{N}$ be infinite and $d_A(n)$ count the number of $a\in A$ which divide $n$ . Is it true that, for every $k$ ,

\limsup_{x\to \infty} \frac{\max_{n<x}d_A(n)}{\left(\sum_{n\in A\cap[1,x)}\frac{1}{n}\right)^k}=\infty?

Source: erdosproblems.com/444

Number theory
Divisors

Wiki page

Status

Solved·, Proved

An accepted solution exists. The statement is true.

Proved. The site labels the problem PROVED; the Erdős–Sárközy theorem that $\max_{n\le x}d_A(n)$ exceeds $\exp(c(\log f_A(x))^2)$ infinitely often, with $f_A(x)$ the reciprocal sum, answers the question yes for every $k$ , as the claim page below records.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 88. Library home: Erdős and Graham 1980.
ErSa80: Erdős, P. and Sárközy, A., Some asymptotic formulas on generalized divisor functions. IV. Studia Sci. Math. Hungar. 15 (1980), 467-479. Library home: Erdős and Sárközy 1980.

Proof claims

Loading proof claims…

Discussion

Loading discussion…

PreviousProblem 443 NextProblem 445

On this page

Statement
Status
References
Proof claims
Discussion

Status: Solved, Proved
Topics: Number theory
Divisors
In the wiki: Wiki page