Problem 462

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Statement. 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 $x$?

Formulation. The site's second sum, unlike its first, does not exclude primes. The formal-conjectures statement, at its commit of 18 September 2026, sums $p(n)/n$ over every $n$ in the window. Terence Tao's comment of 28 September 2025 in the site's discussion thread says that the source is ambiguous on whether primes are excluded: with primes included the question is essentially a weaker form of Legendre's conjecture, and with primes excluded it concerns the semiprimes $pq$ with $p,q=x^{1/2}(\log x)^{O(1)}$ in intervals of length $O(x^{1/2}(\log x)^2)$. This page's standing concerns the site's wording, which includes primes. A prime $n$ contributes $p(n)/n=1$, so an affirmative answer for composites alone gives an affirmative answer as worded.

Status. Open.

Source. erdosproblems.com/462, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #462, https://www.erdosproblems.com/462.

Formalization. Statement in formal-conjectures.

Current assessment

Open; no claim page. The one result recorded is X. Zhang, On the sum of least prime factors in short intervals, arXiv:2608.24930, submitted 22 August 2026 and linked from the site's discussion thread on 18 September 2026. No journal publication or outside review of it is recorded, and the preprint treats composite $n$ only. Unconditionally, it claims $c=8$ in the premise's asymptotic (Theorem 1.1). For each fixed $C>0$, the composite window sums $\mu_C(x)$ over $x\le n\le x+Cx^{1/2}(\log x)^2$ average $4C+O_C(1/\log X)$ over $x\le X$ (Theorem 1.3), with mean-square deviation from $4C$ of $O_C((\log X)^{-2})$ (Theorem 1.4); hence $|\mu_C(x)-4C|\le\varepsilon$ for all but $o(X)$ of the $x\le X$ (Corollary 1.5). These results settle no instance of a question about every large $x$.

Theorem 1.8 gives $\mu_C(x)=4C+O_C(1/\log x)$ for all large $x$, an affirmative answer, under the preprint's Hypothesis 1.6: for every $C_0>0$ and $A\ge1$ there is a constant $C_A$ with $|\pi(y+h)-\pi(y)-h/\log y|\le C_Ah/(\log y)^A$ for all $y\ge2$ and all $h$ with $C_0(\log y)^2\le h\le y$. The proof (Section 5) applies it with $A=2$ at $h$ down to about $C(\log x)^2$. The hypothesis is false. At $C_0=1$, $A=2$ and $h=(\log y)^3$ it bounds $|\pi(y+(\log y)^3)-\pi(y)-(\log y)^2|$ by $C_2\log y$, while Maier's theorem (H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), 221--225, stated for every fixed $N>2$ in Granville's survey) gives, for $N=3$, a $\delta_3>0$ and arbitrarily large $y$ with $\pi(y+(\log y)^3)-\pi(y)>(1+\delta_3)(\log y)^2$. Restricting the hypothesis to the $A=2$ instance the proof uses does not avoid this. So Theorem 1.8 decides nothing, and the preprint has no claim page; the problem has no claim.

Search scope (2026-10-07): the site's page and its discussion thread (two comments, 2025-09-28 and 2026-09-18), the formal-conjectures statement file at its 2026-09-18 commit (no formal proof recorded) and the arXiv record of the preprint.