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Claim. For every $\epsilon$ with $0<\epsilon\le1$ there is a constant $c(\epsilon)$ such that the eventual-time thresholds of the Legendre-symbol partial sums have mean value $c(\epsilon)$ over the primes: $\pi(x)^{-1}\sum_{p\le x}g(\epsilon,p)\to c(\epsilon)$ as $x\to\infty$, where $g(\epsilon,p)$ is the least $t$ such that $\bigl|\sum_{n\le m}(n/p)\bigr|<\epsilon m$ for every $m\ge t$. The proof ends with the quantitative form $\sum_{p\le x}g(\epsilon,p)=c(\epsilon)\pi(x)(1+O((\log\log x)^{-1/8}))$ and identifies $c(\epsilon)=\sum_{\mu\ge1}\mu d_\mu$, where $d_\mu$ is the limiting frequency of the primes with $g(\epsilon,p)=\mu$. Since $\pi(x)\sim x/\log x$, this is the asymptotic $\sum_{p<x}f_\epsilon(p)\sim c_\epsilon x/\log x$ asked by Problem 981, Erdős's display (80) of 1965, with the two-sided threshold $g$ in place of the one-sided threshold $f$ of Erdős and of the problem's statement, and with $c(\epsilon)\ge1$ because every $g(\epsilon,p)\ge1$. The theorem is quoted from the printed page on the result page theorem_p165; the digest is on the source card elliott_1969_conjecture_erdos_concerning_character_sums.

The one-sided threshold. The paper opens with Erdős's one-sided threshold $f(\epsilon,p)$ and the conjecture as its display (1), notes $f(1,p)=n_2(p)$, the least quadratic nonresidue, and then proves the theorem for the two-sided $g(\epsilon,p)$, saying of the original definition only that simple changes in the argument give a similar result (p. 164); no adaptation is printed. Termwise $f(\epsilon,p)\le g(\epsilon,p)$, which does not by itself transfer the asymptotic. The problem's wording, the one-sided form, therefore rests on the printed theorem together with the author's remark, as the problem page's Status records. For $\epsilon\ge1$ the one-sided form is settled without this paper: the problem page's Formulation shows that for $\epsilon>1$ both thresholds equal $1$ for every odd prime, and records the identity $f(1,p)=n_2(p)$ that makes the instance $\epsilon=1$ Erdős's theorem (78) under its Origin.

Argument, in outline. The primes $p\le x$ whose two-sided sums reach $\epsilon m$ for some $m\ge N$ are few by a fourth-moment large-sieve bound (Lemma 7); for the rest, $g(\epsilon,p)=\mu\le N$ is decided by the symbols $(n/p)$ for $n\le N$, so by the residue of $p$ modulo $N!$, and with $N=[\sqrt{\log\log x}]$ the Siegel--Walfisz theorem gives each $\mu$ its limiting frequency $d_\mu$. The proof was not checked here beyond this outline.

Depends on. Nothing in this wiki.

Acceptance. Refereed: P. D. T. A. Elliott, A conjecture of Erdős concerning character sums, Indagationes Mathematicae (Proceedings) 72 (1969), no. 2, 164--171 (Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31), communicated by N. G. de Bruijn at the meeting of 25 January 1969, the date this page is named by. Reviewed: the site's curator, Thomas F. Bloom, marks the problem PROVED and credits the proof to this paper in the problem's commentary (page last edited 27 December 2025); the curator neither wrote nor submitted the result. Independently of the site, Tang and Zhang, who found the paper while writing on the first-passage variant, restate Erdős's (80) as their Conjecture 1.1 and write that Elliott proved it (arXiv:2512.24631v2, p. 2; card tang_2025_average_first_passage_times_character_sums), and the site's thread of 27 December 2025 records the same reading. Not formalized: no Lean statement of the problem exists, and nothing was built or audited here. The one-sided qualification above is disclosed and is not a dispute; no dispute of the theorem was found in the searches dated on the problem page.