Tao 2023 convergence alternating series erdos assuming hardy

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Terence 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, published online 2024-02-01; arXiv:2308.07205 (2023). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2308.07205), every other right reserved.

Erdos asked (problem 15) whether the alternating series sum_{n>=1} (-1)^n n / p_n converges, where p_n is the nth prime; numerical computation (p. 1) suggests slow convergence to roughly -0.052161. Section 2 records Said's unpublished equivalence with the convergence of sum_{n>=2} (-1)^{pi(n)} / (n log n), so the question is really about how the parity of the prime counting function is distributed. Theorem 1.4, the paper's main result, proves that both series converge assuming Conjecture 1.3, a quantitative form of the Hardy-Littlewood prime tuples conjecture with power-saving error uniform over k <= (log log x)^5 and shifts in [0, log^2 x], a very slightly strengthened form of V. Kuperberg's Conjecture 1.3 (k <= (log log x)^3 there). The proof (outlined on p. 2) first applies the van der Corput A-process, which reduces matters to short windows (x, x + lambda log x] with lambda of order (log log x)^{4.4}: the number of primes in such a window must be odd about as often as even. Kuperberg's mean-value bounds for singular series do not reach the large k this needs, so Tao works instead with the Banks-Ford-Tao random sifted model, computing probabilities on the model with cancellation across tuple sizes k; the mechanism is that many sifting steps smooth out, rather than amplify, parity bias in the number of survivors. The result is thus a conditional affirmative answer to problem 15.

Conjecture 1.3 here is Tao's restatement of Kuperberg's Conjecture 1.3 (Q. J. Math. 74 (2023); arXiv:2210.09775v2), filed as kuperberg_2023_sums_singular_series_large_sets_tail / conjecture_1_3, with the tuple-size range widened from (log log x)^3 to (log log x)^5, the restriction x >= 10 added, and the admissibility requirement dropped (p. 2). The 2026 conditional claims on problem 251 assume Kuperberg's original form; the Kuperberg card and that problem page record them, and this card's account of problem 15 is unchanged.

Source: https://arxiv.org/abs/2308.07205.

The copy read for this card is the arXiv v3 of 23 August 2023 (365,554 bytes), whose abstract page carries no journal reference. The journal publication above is taken from the Crossref record of DOI 10.1090/cams/29; the published text was not compared with the arXiv version.

Bears on. #15

Results to transcribe.

• Theorem 1.4: Assuming the quantitative Hardy-Littlewood prime tuples Conjecture 1.3, sum (-1)^{pi(n)}/(n log n) converges, and hence so does Erdos's series sum (-1)^n n/p_n.
• Question 1.1 / 1.2 equivalence: Convergence of sum_{n>=1} (-1)^n n/p_n is equivalent to convergence of sum_{n>=2} (-1)^{pi(n)}/(n log n) (an observation of Said; proved in Section 2).
• Conjecture 1.3: The quantitative prime tuples hypothesis used: for k <= (log log x)^5 and shifts in [0, log^2 x], the prime k-tuple count matches the singular-series prediction with error O(x^{1-eps}).
• Method: Van der Corput A-process reduces to equidistribution of the parity of pi(x + lambda log x) - pi(x) for lambda ~ (log log x)^{4.4}, handled via the Banks-Ford-Tao random sifted model of the primes.
• Numerics: Computed partial sums settle slowly near -0.052161 (p. 1); Erdos noted the variant sum (-1)^n n log n / p_n diverges since its terms do not tend to zero.

The author-recorded reconstruction of Theorem 1.4 and its same-paper inputs is filed under the research folder Alternating series of n over the nth prime, on the pages Theorem 1.4 (Section 3, pp. 4--12, with Conjecture 1.3 as the imported hypothesis), Lemma 3.1 (p. 6), Lemma 3.2 (pp. 7--10, with the random sifted model) and relation (2.1) (Section 2, pp. 3--4, the equivalence of Questions 1.1 and 1.2). Those pages consume this card; no result pages are extracted here, and the reconstruction is not an independent review.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.