Does the alternating prime series $\sum(-1)^n n/p_n$ converge? (Erdős #15)
Statement
Let $p_n$ denote the $n$th prime. Is it true that the series $$\sum_{n=1}^{\infty}(-1)^n\frac{n}{p_n}$$ converges? By the prime number theorem $p_n\sim n\log n$, so the terms $n/p_n\sim 1/\log n$ tend to $0$; but they are not monotonic, so convergence does not follow from the alternating series test and is genuinely in question.
Acceptance. FULLY RESOLVES: an unconditional proof that $\sum_{n\ge1}(-1)^n n/p_n$ converges, or an unconditional proof that it diverges — machine-checkable (Lean/Coq) preferred, otherwise a complete written proof. A proof relying on an unproven hypothesis (such as Hardy–Littlewood) must state the hypothesis explicitly and counts only as an ADVANCES. ADVANCES (each a complete proof, strictly beyond what is stated in the background): a proof of convergence of $\sum_{n}(-1)^n n/p_n$ under a hypothesis strictly weaker than the strong Hardy–Littlewood assumption used by Tao; or an unconditional proof of one of the companion claims — that $\sum_{n}(-1)^n\frac{1}{n(p_{n+1}-p_n)}$ converges, or that $\sum_{n}(-1)^n\frac{1}{n(p_{n+1}-p_n)(\log\log n)^c}$ converges for some $c\le 2$ (the best proven is absolute convergence only for $c>2$). Deliver a machine-checkable or complete written proof.
Background
Posed by Erdős [Er97, p.158], [Er97e, p.535], [Er98]; listed as open on erdosproblems.com/15 (fetched 2026-07-21, status 'open'). Erdős suggested that a computer could be used to explore the series and saw no other method of attack. Tao [Ta23] proved that the series converges assuming a strong form of the Hardy–Littlewood prime $k$-tuples conjecture. In [Er98] Erdős posed companion questions: he conjectured that $\sum_{n}(-1)^n\frac{1}{n(p_{n+1}-p_n)}$ converges while $\sum_{n}(-1)^n\frac{1}{p_{n+1}-p_n}$ diverges — Weisenberg observes that the existence of infinitely many bounded prime gaps (Zhang, 2014) already shows the latter series fails to converge, and gives a further argument that, assuming Hardy–Littlewood, this latter series is unbounded in at least one direction (positive or negative). Erdős further conjectured that $\sum_{n}(-1)^n\frac{1}{n(p_{n+1}-p_n)(\log\log n)^c}$ converges for every $c>0$; he and Nathanson proved it converges absolutely for $c>2$ (and, conditional on hard prime-gap conjectures, not absolutely for $c=2$), and Sawhney gave a Selberg-sieve proof of the $c>2$ absolute convergence. The statement has a Lean formalization in the formal-conjectures project; no Erdős prize is attached. Attacker's tool: high-precision numerical evaluation of partial sums to probe the oscillation, combined with sieve and analytic estimates for the distribution of prime gaps to control the auxiliary series.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #15 (T. F. Bloom) | website |
| REF-02 | Lean formalisation (formal-conjectures, Erdős #15) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.