Maynard 2020 primes arithmetic progressions large moduli iii

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corollary_1_4: Maynard's corollary that for delta > 0 sufficiently small, outside a set of at most x^(1/2+delta)/(log x)^A moduli, every q <= x^(1/2+delta) with a divisor in [x^(2/5+delta), x^(3/7)] has pi(x,a;q) of the order pi(x)/phi(q) for every a coprime to q.

theorem_1_1: Maynard's theorem that, for moduli q_1 q_2 with q_1 near Q_1 <= x^(1/10-3delta)/(log x)^C and q_2 near Q_2 <= x^(4/10+4delta)(log x)^C, the sum over the moduli of the largest discrepancy over all primitive residue classes is O_C(delta pi(x) + x(log log x)^2/(log x)^2).

theorem_1_2: Maynard's theorem that for 0 < delta < 1/1000 and Q_1 Q_2 Q_3 = x^(1/2+delta) in a stated range, primes are equidistributed with error O(x/(log x)^A) on average over moduli q_1 q_2 q_3, uniformly over residue classes whose class modulo q_1 q_2 does not depend on q_3.

theorem_1_3: Maynard's theorem that for delta > 0 sufficiently small there is a minorant rho of the prime indicator with sum up to x at least pi(x)/8 that is equidistributed with error O(x/(log x)^A), uniformly over primitive residue classes, on average over moduli q_1 q_2 with q_1 <= Q_1 in [x^(2/5+5delta), x^(3/7)] and q_2 <= x^(1/2+delta)/Q_1.

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James Maynard, Primes in arithmetic progressions to large moduli III: Uniform residue classes. Mem. Amer. Math. Soc. 306 (2025), no. 1544. DOI: 10.1090/memo/1544. arXiv:2006.08250. The copy read for this card is arXiv:2006.08250v1 (15 Jun 2020). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2006.08250), every other right reserved.

The paper extends Bombieri-Vinogradov to moduli of size x^{1/2+delta} with conveniently sized divisors, with estimates that are completely uniform over residue classes, unlike the earlier Bombieri-Fouvry-Friedlander-Iwaniec results tied to a single fixed class a. Theorem 1.1 gives a uniform equidistribution statement with a weak error term for moduli q_1 q_2 with Q_1 <= x^{1/10-3delta}/(log x)^C and Q_2 <= x^{4/10+4delta}(log x)^C, bounding the sum of suprema over residue classes by O_C(delta pi(x) + x(log log x)^2/(log x)^2). Theorem 1.2 gives almost uniform equidistribution for moduli factoring as Q_1 Q_2 Q_3 = x^{1/2+delta} with explicit constraints on Q_2, Q_3, and Theorem 1.3 constructs, for delta sufficiently small, a prime minorant rho with uniform equidistribution and sum_{n<=x} rho(n) >= pi(x)/8; Corollary 1.4 then shows that outside a sparse bad set, every primitive residue class modulo q <= x^{1/2+delta} with a divisor in [x^{2/5+delta}, x^{3/7}] contains the expected order of primes. The technique combines Type II estimates, a de-amplifying refinement requiring three conveniently sized factors, and triple divisor function bounds; the constants are ineffective because of possible Siegel zeros.

Source: https://arxiv.org/abs/2006.08250.

Bears on. #158: the paper does not mention the problem. Its results count primes in single progressions to moduli up to x^{1/2+delta} with delta small, and none bounds a set with few representations as a sum of two elements.

Results. Labels and pages are those of v1. Theorem 1.1 (p. 3, uniform equidistribution with a weak error term); Theorem 1.2 (p. 3, almost uniform equidistribution to three-factor moduli); Theorem 1.3 (p. 4, an equidistributed minorant for the primes); Corollary 1.4 (p. 4, primes in every primitive class for almost all suitably factored moduli). The two remarks on p. 4, on ineffective constants and on improving the error terms of Theorems 1.2 and 1.3 by excluding bad moduli, are recorded on the theorem pages. Propositions 5.1-5.4 (pp. 8-10), the Type II and ternary-divisor estimates the theorems are deduced from, and Corollary 5.5 (p. 10) on the ternary divisor function are proof inputs and have no pages.

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