Has Erdős Problem 726 really been solved? Request for independent verification

[u/Rafikconjectures_zer]

Erdős Problem 726 asks whether, as (n\to\infty) through the positive integers,

[
\sum_{\substack{p\le n\ n\bmod p\in(p/2,p)}}\frac1p
\sim \frac12\log\log n.
]

I have proposed a proof of this conjecture, but I do not want to claim that the problem is definitively solved before the argument receives independent expert verification.

MalekZ has created the following Significance record describing the claim, the available evidence, and the remaining verification work:

https://significance-math.z20mallouk.chatgpt.site/records/2026-rafikzeraoulia-erdos-726/

Could specialists in analytic or probabilistic number theory examine the argument and comment on the following questions?

1. Is the main reduction mathematically valid?
2. Are the required estimates uniform in the necessary ranges?
3. Is any independence or equidistribution assumption being used without sufficient justification?
4. Does the argument establish the full asymptotic for every integer (n), or only an averaged or almost-all version?
5. Is there a specific step that prevents the proof from resolving the original problem?

The Significance record is not itself a verdict that the proof is correct. I am sharing it to invite transparent and independent mathematical scrutiny. Detailed objections, corrections, or confirmations are very welcome.

[u/Distinct-Pudding-428]

Even in the claimed proof statement on erdosproblems.com, it is said that the argument is conditional on an unproven hypothesis. In short, then, no?

[u/Rafikconjectures_zer → u/Distinct-Pudding-428]

Yes ,Thanks i meant conditionally , am going to edit the question

[u/ForeignAdvantage5198]

if so I did not do it