Zeraoulia (2026): A conditional resolution of an Erdős–Graham–Ruzsa–Straus conjecture on reciprocal sums over primes

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Rafik Zeraoulia, A Conditional Resolution of an Erdős–Graham–Ruzsa–Straus Conjecture on Reciprocal Sums over Primes, Zenodo, preprint, version 1.0, issued 11 August 2026, CC BY 4.0, doi:10.5281/zenodo.21882603 (concept DOI 10.5281/zenodo.21882602).

The record's file EGRS_Conditional_Resolution.pdf was not retrieved: zenodo.org answered the record page, its API and the file download with HTTP 403 ("unusual traffic") on 2026-09-27, so its size is not recorded and no copy was read. The bibliographic data above are the DataCite metadata of the DOI.

Scope, from the record's abstract and the erdosproblems.com proof-claims thread: for $S(n)=\sum 1/p$ over the primes $p\le n$ with $n\bmod p>p/2$, the sum of Problem 726, the preprint claims $S(n)=\tfrac12\log\log n+O(\log\log\log n)$ conditionally on a "Reciprocal-Prime Equidistribution Hypothesis", an extension of the smooth equidistribution estimate over primes, Proposition 1.12 of Matomäki, Radziwiłł, Shao, Tao and Teräväinen, to the range $|N|\le\exp(P^c)$ for primes $p\asymp P$. The abstract states that the hypothesis is unproved and that no unconditional resolution is claimed. The preprint was posted on the site's proof-claims thread for Problem 726 on 11 August 2026 as a full proof, with the submitter's own note that the result is conditional; a comment of 12 August 2026 on that thread classes it as partial, and a third-party evidence record of 16 August 2026, linked from the thread on 24 August 2026, lists no independent rerun, assessment or review. No refereed publication and no independent review were found on 2026-09-27. A conditional theorem does not resolve the problem, whatever the standing of the argument.

Source: https://doi.org/10.5281/zenodo.21882603.

Bears on. #726

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