A Conditional Resolution of an Erdős–Graham–Ruzsa–Straus Conjecture on Reciprocal Sums over Primes

Erdős, Graham, Ruzsa and Straus conjectured in 1975 that the reciprocal sum over primes p ≤ n satisfying n mod p > p/2 is asymptotic to (1/2) log log n. This manuscript gives a complete conditional proof under an explicitly stated Reciprocal-Prime Equidistribution Hypothesis: a one-dimensional exp(P^c)-range extension of the smooth equidistribution estimate of Matomäki, Radziwiłł, Shao, Tao and Teräväinen. Under this hypothesis, the stronger estimate S(n) = (1/2) log log n + O(log log log n) is obtained by smoothing the half-interval indicator, applying dyadic reciprocal-prime equidistribution, and exploiting cancellation in the corresponding continuous integral. The hypothesis is currently unproved, and no unconditional resolution is claimed.