Erdos's Question on Shifted Pairwise-Coprime Reciprocal Sums
Prior state unknown→proved
the average order; the uniform bound Erdos asked about is not settled
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theory / Number theory
Erdos's Question on Shifted Pairwise-Coprime Reciprocal Sums
Let $\mathcal{M}(n)$ be the supremum of $\sum_{a \in A} 1/(n-a)$ over pairwise coprime $A \subset [1,n)$. Erdos asked whether $\mathcal{M}(n) \le \sum_{p<n} 1/p + O(1)$ uniformly. The average order is settled: $\sum_{n \le N} \mathcal{M}(n) = e^{-\gamma} N \log\log N + O(N)$.
10Significance / 100
1Frontier events
0Verification tasks
0Recorded attempts
Temporal state
Current frontier
No reconciled state yet.
Append-only history
Frontier timeline
Research memory
Claims and attempts
Scoped claims
Let $\mathcal{M}(n)$ be the supremum of $\sum_{a \in A} 1/(n-a)$ over pairwise coprime $A \subset [1,n)$. Erdos asked whether $\mathcal{M}(n) \le \sum_{p<n} 1/p + O(1)$ uniformly. The average order is settled: $\sum_{n \le N} \mathcal{M}(n) = e^{-\gamma} N \log\log N + O(N)$.
the average order; the uniform bound Erdos asked about is not settled
Recorded attempts
Evidence graph
Connected research record
No public relationships recorded yet.