Erdős Problem #462

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

The least prime factor $p(n)$ is defined for $n\geq2$. The nonprime terms in the first sum are composite integers, omitting the undefined $n=1$ term. The second sum, as written in the source, is not restricted to composites. [S1] [S2]

Statement recorded in the supplied source.

Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that $ \sum_{\substack{n<x\\ n\textrm{ not prime}}}\frac{p(n)}{n}\sim c\frac{x^{1/2}}{(\log x)^2}. $ Is it true that there exists a constant $C>0$ such that $ \sum_{x\leq n\leq x+Cx^{1/2}(\log x)^2}\frac{p(n)}{n} \gg 1 $ for all large $x$? [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

What are the global asymptotics and short-interval lower bounds for sums of the least prime factor divided by the integer itself?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2130 (EP-462), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 462, https://www.erdosproblems.com/462. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Paul Erdős and Ronald Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 92. (source_collection). https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf. Bibliographic information from the supplied record.