Grosswald 1982 arithmetic progressions that consist only primes

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corollary_p12: States the paper's unnumbered Corollary: if the asymptotic formula (2) for the number of m-term prime progressions up to x holds, then there are arbitrarily long arithmetic progressions consisting only of primes.

theorem_1: States that the Strong Theorem X_1, a form of Hardy and Littlewood's conjectural Theorem X_1 made uniform in an auxiliary parameter, would imply an explicit asymptotic formula, of order x^2/(log x)^m, for the number of m-term arithmetic progressions of primes up to x.

theorem_2: Proves unconditionally that the number of three-term arithmetic progressions of primes up to x is (C/2) x^2/(log x)^3 times an asymptotic series in 1/log x with computable coefficients, C the twin prime constant, a_1 = 7/2 - log 2 and a_2 given in closed form.

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Emil Grosswald, Arithmetic progressions that consist only of primes. Journal of Number Theory 14 (1982), no. 1, 9-31. doi:10.1016/0022-314X(82)90055-5. The copy read for this card is the publisher's scan of the article, printed pp. 9-31. It prints "0022-314X/82/010009-23$02.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved." in the footer of p. 9, every other right reserved.

Let N_m(x) count the arithmetic progressions of m primes 3 <= p_1 < ... < p_m with p_m <= x (pp. 9-10), and let F_m(x), of the form C_m x^2/log^m x with C_m explicit in the abstract's notation, be the right-hand side of the paper's formula (2) (p. 11). Theorem 1 (p. 11) shows that the "Strong Theorem X_1", a form of Hardy and Littlewood's conjectural Theorem X_1 (the prime k-tuple conjecture) made uniform in an auxiliary parameter, which the paper notes they neither proved nor claimed, would imply N_m(x) ~ F_m(x). The abstract states the conditional result as the series N_m(x) = F_m(x){1 + sum_{j=1}^N a_j log^{-j} x + O((log x)^{-N-1})} with explicitly computable coefficients; in the body that refinement, (2'), is credited to Zagier and, like (2) itself, obtained only heuristically (p. 11), and the concluding remarks (p. 29) call the series formulae for general m, built from Lemma 3 (p. 25), no more than conjectures. Theorem 2 (p. 12) proves the series unconditionally for m = 3, using the Vinogradov form of the Hardy-Ramanujan-Littlewood circle method; the leading term is (C/2) x^2/log^3 x with C the twin-prime constant, and the first coefficients are computed in closed form (a_1 = 7/2 - log 2 = 2.8068528194... and a_2 = 13 - 5 log 2 - log^2 2 - pi^2/12 = 8.23134404...). The abstract calls the cases m = 1 and m = 2 rather trivial, and the body notes that N_2(x) equals pi(x)^2/2 (p. 12). As an immediate corollary of Theorem 2, N_3(x) tends to infinity (p. 10); the paper notes that this was implicit in a theorem of van der Corput and stated explicitly by Chowla, and that a lower bound N_3(x) >= C_0 x^2/log^3 x with some C_0 > 0 is almost immediate from Estermann's work. An unnumbered Corollary (p. 12) observes that if (2) holds, there are arbitrarily long arithmetic progressions of primes, in connection with "the conjecture stated in the Introduction". The Introduction (p. 9) states the old conjecture that such progressions exist and recalls the stronger one, attributed "presumably first" to Erdős, that a set of integers whose reciprocals have a divergent sum contains arbitrarily long arithmetic progressions. Section 9 (pp. 29-30) compares N_3(x) with the predicted values for x up to 50,000 (Table I, p. 30).

Source: https://doi.org/10.1016/0022-314X(82)90055-5.

Read status: claims checked for Theorem 1 (p. 11), Theorem 2 (p. 12) and the Corollary (p. 12), each read clause by clause on the print; the derivation of Theorem 1 (Section 3, pp. 13-15) and the proof of Theorem 2 (Sections 4-7, pp. 15-25) were read for their structure only, and Lemma 3 (p. 25) and its sketched proof (pp. 26-29) as statements. Nothing is independently reviewed. Result pages: theorem_1, theorem_2 and corollary_p12.

Bears on. #200: background only. The paper's counts concern progressions with a fixed number m of terms, unconditional for m = 3 (Theorem 2) and conditional for larger m (Theorem 1), and the Corollary gives, conditionally, progressions of every fixed length; none of them treats progressions whose length grows with N, which the problem asks about. #3: background only. The paper recalls the problem's conjecture on p. 9, and its Corollary concerns only the primes, one set with a divergent sum of reciprocals, conditionally on the unproved formula (2).

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