Lichtman 2024 modification linear sieve count twin primes

../

proposition_5_4: Lichtman's proposition that Iwaniec's well-factorable linear-sieve weights equidistribute primes in a fixed residue class over moduli with prescribed large prime factors up to a level theta(t_1) that depends on the size x^(t_1) of the largest of them, and to (3 - u)/5 over smooth moduli.

theorem_1_1: Lichtman's theorem that at level D = x^(10/17 - eps) there are sieve weights in {-1, 0, 1} that equidistribute primes in a fixed residue class on average over moduli d <= D and still give a linear-sieve upper bound whose main-term function is at most 1.000081 F(s) for 1 <= s <= 3.

theorem_1_2: Lichtman's theorem that the number of twin primes up to x is asymptotically at most 3.29956 Pi(x), where Pi(x) is the Hardy-Littlewood prediction, a 2.94% improvement on Wu's bound 3.39951.

theorem_2_12: Lichtman's main technical theorem that at level D = x^(7/12 + eta) the linear-sieve upper bound holds with main-term function F*(s) and a remainder weighted by a sum of programmably factorable sequences, where F*(s) = F(s) + O(eta^5) for eta < 1/204.

---

Jared Duker Lichtman, A modification of the linear sieve, and the count of twin primes. Algebra & Number Theory 19 (2025), no. 1, 1-38. doi:10.2140/ant.2025.19.1. arXiv:2109.02851. The copy read for this card is arXiv:2109.02851v2 (14 February 2024); the arXiv record names arXiv's non-exclusive distribution license (arXiv:2109.02851), every other right reserved.

Theorem 1.1 (pp. 2-3) constructs sieve weights lambda*(d) in {-1, 0, 1} whose strong factorization properties give, for any fixed residue a and any A, eps > 0, the equidistribution estimate sum over d <= D with (d, a) = 1 of lambda*(d) (pi(x; d, a) - pi(x)/phi(d)) <<_{a,A,eps} x/(log x)^A at level D = x^{10/17 - eps}, beyond the x^{4/7} of Bombieri-Friedlander-Iwaniec for well-factorable weights and Maynard's x^{7/12} for Iwaniec's linear-sieve weights, while the weights still give a linear-sieve upper bound with main-term function F*(s) <= 1.000081 F(s) for 1 <= s <= 3. Its full technical form is Theorem 2.12 (pp. 8-9): at level D = x^{7/12 + eta} the weights are a sum of at most exp(eps^{-3}) programmably factorable sequences and F*(s) = F(s) + O(eta^5) for eta < 1/204, and 7/12 + 1/204 = 10/17. The key idea (p. 4) is that up to level x^{10/17} the integers in the linear sieve's support that fail Maynard's factorization conditions form two explicit families contributing O(eta^5), so the weights can be revised on those few d. Theorem 1.2 (p. 3) gives pi_2(x) <~ 3.29956 Pi(x), which the paper calls a 2.94% improvement on Wu's 2004 bound 3.39951 and the largest percentage gain since Bombieri-Friedlander-Iwaniec in 1986. Section 6 opens by saying it applies the modified sieve (p. 26), but the remainder estimates its proof invokes are the variable-level Proposition 5.4 and Corollary 5.6 for Iwaniec's weights, together with Wu's iteration; it does not invoke Theorem 2.12 or the modified weights of Theorem 1.1.

Source: https://arxiv.org/abs/2109.02851.

Read status. Claims checked: Theorems 1.1, 1.2 and 2.12, Corollary 2.13 and Proposition 5.4 were read clause by clause on the printed pages of arXiv:2109.02851v2. The proofs were read in outline only; the numerical computations were not checked.

Bears on. #158: the problem page names this paper only in its list of linked library material, which is generated from this card's own link. The paper concerns primes in arithmetic progressions and does not mention sets with bounded representation functions. Its equidistribution estimates average over moduli with signed sieve weights for one fixed residue class and give no bound for a single modulus or uniformly over residues. It settles no part of Problem 158.

Results. Theorem 1.1 (pp. 2-3); Theorem 1.2 (p. 3); Theorem 2.12 (pp. 8-9, with Definition 2.4 and Corollary 2.13); Proposition 5.4 (p. 24).

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