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Claim. Assume Conjecture 1.3 of the paper, a quantitative form of the Hardy-Littlewood prime tuples conjecture. Then the series

$$\sum_{n\ge2}\frac{(-1)^{\pi(n)}}{n\log n}$$

converges, and therefore so does the series $\sum_{n\ge1}(-1)^n n/p_n$ of Problem 15. This is Theorem 1.4 of Tao's paper. The two series stand or fall together by an observation of Said that Section 2 of the paper proves as relation (2.1): the sign $(-1)^{\pi(n)}$ changes exactly at the primes, and the partial sums of $\sum(-1)^nn/p_n$ differ from half of the partial sums of the second series, taken up to $x\log x$, by a quantity that converges (a constant plus $o(1)$); the reconstruction of that relation is filed at relation (2.1). The proof of the theorem applies the van der Corput $A$-process, which reduces convergence to the equidistribution of the parity of the number of primes in short intervals $(x,x+\lambda\log x]$ with $\lambda$ of order $(\log\log x)^{4.4}$, and then replaces the primes by the random sifted model of Banks, Ford and Tao, in which the sifting steps damp the irregularities of that parity; the cancellation between tuples of different sizes is what the uniform range of the hypothesis supplies.

Hypothesis. Conjecture 1.3 asks that, for $x\ge10$, every tuple size $k\le(\log\log x)^5$ and every choice of $k$ distinct shifts in $[0,\log^2x]$, the number of $n\le x$ for which all $k$ shifted values are prime equals the singular-series prediction of Hardy and Littlewood up to a power-saving error, uniformly in $k$ and in the shifts. It is Tao's restatement of a conjecture of Kuperberg, filed as Kuperberg's Conjecture 1.3, with the tuple-size range widened from $(\log\log x)^3$ to $(\log\log x)^5$ and the admissibility requirement dropped. The hypothesis is unproved, and the claim gives no unconditional answer.

Scope. The claim is conditional and settles no standing of the problem by itself: unconditionally, whether the partial sums of $\sum(-1)^nn/p_n$ converge remains open. The absolute series $\sum n/p_n$ diverges, since $n/p_n\sim1/\log n$, so any convergence is conditional in the analytic sense as well. The paper's numerical computation (p. 1) suggests slow convergence to roughly $-0.052161$.

The library card Tao 2023 records the paper in its arXiv v3 text; the published text has not been compared with it. A reconstruction of the theorem and its same-paper lemmas is filed under the Problem 15 research folder; its four pages were each independently reviewed in a focused review on 2026-09-28, with the verdicts fidelity faithful and the Theorem 1.4 argument sound conditional on Conjecture 1.3, and no tier assigned. Those reviews cover the reconstruction and are not acceptance evidence for this claim, whose only listed evidence is the refereed publication; the paper's own text has not been independently reviewed here.

Acceptance. The result is refereed: T. Tao, The convergence of an alternating series of Erdős, assuming the Hardy--Littlewood prime tuples conjecture, Comm. Amer. Math. Soc. 4 (2024), no. 3, 80--96, DOI 10.1090/cams/29. The site's commentary records that the paper proves convergence under a strong form of the prime tuples conjecture and keeps the problem open, so no curator acceptance of a solution is listed. The page is dated by the arXiv posting of 14 August 2023.

Depends on. Nothing on this wiki beyond the cited paper; its hypothesis, Conjecture 1.3, is stated above.