Maynard 2020 primes arithmetic progressions large moduli i

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corollary_1_2: For 0 < delta < 1/42 and 0 < eta < (1-42 delta)/4, the absolute errors for primes in a fixed class a, summed over moduli q <= x^(1/2+delta) coprime to a that have a divisor in an explicit window, are O(x/(log x)^A).

corollary_1_3: For 0 < delta < 1/55, A > 0 and Q <= x^(1/2+delta), all but at most 18 delta Q phi(a)/a moduli q in [Q, 2Q] coprime to a satisfy pi(x;q,a) = (1 + O((log x)^(-A))) pi(x)/phi(q).

corollary_1_4: For a fixed integer a and eps > 0, the absolute errors for primes in the class a, summed over q_1 <= x^(1/21) and q_2 <= x^(10/21-eps) both coprime to a with modulus q_1 q_2, are O(x/(log x)^A) for every A > 0.

theorem_1_1: For a fixed integer a and moduli q_1 q_2 with q_1 <= Q_1 and q_2 <= Q_2 coprime to a, the absolute errors in the prime count in the class a sum to O(x/(log x)^A) whenever Q_1 Q_2^2, Q_1^12 Q_2^7 and Q_1^20 Q_2^19 lie below x^(1-100 eps), x^(4-100 eps) and x^(10-100 eps).

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James Maynard, Primes in arithmetic progressions to large moduli I: Fixed residue classes. Mem. Amer. Math. Soc. 306 (2025), no. 1542. DOI: 10.1090/memo/1542. arXiv:2006.06572. The copy read for this card is arXiv:2006.06572v2 (5 Apr 2021). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2006.06572), every other right reserved.

Theorem 1.1 bounds the sum of absolute errors |pi(x; q, a) - pi(x)/phi(q)| over moduli q = q_1 q_2 with q_1 <= Q_1, q_2 <= Q_2 and (q_1 q_2, a) = 1, subject to the size constraints Q_1 Q_2^2 < x^{1-100 epsilon}, Q_1^{12} Q_2^7 < x^{4-100 epsilon} and Q_1^{20} Q_2^{19} < x^{10-100 epsilon}, for a fixed integer residue a. Corollaries 1.2 and 1.4 deduce Bombieri-Vinogradov-type bounds for moduli q <= x^{1/2+delta} possessing a divisor in a prescribed range and for products q_1 q_2 with q_1 <= x^{1/21} and q_2 <= x^{10/21-epsilon}, reaching moduli up to x^{11/21-epsilon}; Corollary 1.3 shows that for Q <= x^{1/2+delta} with 0 < delta < 1/55 all but at most 18 delta Q phi(a)/a moduli q in [Q, 2Q] coprime to a admit the expected prime count, so for instance 99% of moduli in [Q, 2Q] coprime to a are fine when Q <= x^{1/2+1/2000}. The method extends the Bombieri-Fouvry-Friedlander-Iwaniec circle of techniques with amplification-inspired ideas and Zhang/Polymath refinements, ultimately using Kuznetsov trace formula bounds for sums of Kloosterman sums and Weil/Deligne style algebraic-geometry estimates.

Source: https://arxiv.org/abs/2006.06572.

Read status. Claims checked: Theorem 1.1 and Corollaries 1.2-1.4 (pp. 3-4) and the deductions of the corollaries from Theorem 1.1 (Section 4, pp. 9-10) were read clause by clause on the printed pages. The proof of Theorem 1.1 (Sections 7-20, pp. 12-101) was read for structure only.

Bears on. #158: the paper does not mention the problem, and Theorem 1.1, as an average over moduli for one fixed residue class, gives no bound for an individual modulus or for residues varying with the modulus.

Results. Labels and pages are those of v2.

• Theorem 1.1 (p. 3): the averaged absolute-error bound over moduli q_1 q_2 under the size conditions (1.3)-(1.5).
• Corollary 1.2 (p. 3): the same bound over moduli q <= x^(1/2+delta) with a divisor in an explicit window, for 0 < delta < 1/42.
• Corollary 1.3 (p. 3): for 0 < delta < 1/55 and Q <= x^(1/2+delta), all but at most 18 delta Q phi(a)/a moduli in [Q, 2Q] coprime to a satisfy pi(x;q,a) = (1 + O_(a,delta,A)((log x)^(-A))) pi(x)/phi(q).
• Corollary 1.4 (p. 4): the bound of Theorem 1.1 for q_1 <= x^(1/21) and q_2 <= x^(10/21-eps), moduli up to x^(11/21-eps).

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