Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Maynard 2020 primes arithmetic progressions large moduli ii
theorem_1_1: For fixed a and A, epsilon > 0, the sum over q <= Q coprime to a of lambda_q (pi(x;q,a) - pi(x)/phi(q)) is O_{a,A,epsilon}(x/(log x)^A) when lambda_q is triply well factorable of level Q <= x^(3/5-epsilon), extending the x^(4/7-epsilon) range of Bombieri, Friedlander and Iwaniec.
theorem_1_2: For fixed a and A, epsilon > 0 and the well-factorable upper bound linear sieve weights lambda^+ of level D <= x^(7/12-epsilon), the sum over q <= x^(7/12-epsilon) coprime to a of lambda_q^+ (pi(x;q,a) - pi(x)/phi(q)) is O_{a,A,epsilon}(x/(log x)^A).
James Maynard, Primes in arithmetic progressions to large moduli II: Well-factorable estimates. Mem. Amer. Math. Soc. 306 (2025), no. 1543. DOI: 10.1090/memo/1543. arXiv:2006.07088. The copy read for this card is arXiv:2006.07088v1 (12 Jun 2020). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2006.07088), every other right reserved.
Theorem 1.1 (p. 2) shows that for an integer a, A, epsilon > 0 and weights lambda_q that are triply well factorable of level Q <= x^{3/5-epsilon}, the sum over q <= Q with (a,q) = 1 of lambda_q (pi(x;q,a) - pi(x)/phi(q)) is O_{a,A,epsilon}(x/(log x)^A), extending the bound of Bombieri, Friedlander and Iwaniec (quoted as Theorem A, p. 2), which handles well-factorable weights of level up to x^{4/7-epsilon}. Theorem 1.2 (p. 3) gives the same bound for the well-factorable upper bound sieve weights lambda^+ of the linear sieve of level D <= x^{7/12-epsilon}, the sum running over q <= x^{7/12-epsilon} with (q,a) = 1; these weights are not triply well factorable of level D, and the proof (Section 9) factors the elements of their support directly (Proposition 9.1, p. 23). Here pi(x) counts the primes less than x. Well factorable of level Q (Definition 1, p. 2) means that for every factorization Q = Q_1 Q_2 with Q_1, Q_2 >= 1 the sequence is a Dirichlet convolution of two sequences bounded by 1 in absolute value and supported on [1, Q_1] and [1, Q_2]; triply well factorable (Definition 2, p. 2) asks the same for every Q = Q_1 Q_2 Q_3 with three such sequences. The proof uses Heath-Brown's identity, Deshouillers-Iwaniec bounds for sums of Kloosterman sums from the Kuznetsov trace formula, and Weil-bound estimates for the divisor function in arithmetic progressions.
Source: https://arxiv.org/abs/2006.07088.
Bears on. #158: the paper does not mention the problem; Theorem 1.2 averages over moduli in one fixed residue class and gives no estimate for a single modulus or varying residues.
Results. Labels and pages are those of v1.
- Theorem 1.1 (p. 2): triply well factorable weights of level Q <= x^{3/5-epsilon} give the error saving x/(log x)^A for primes in a fixed residue class; the page also states Definitions 1 and 2 and the quoted Theorem A (p. 2).
- Theorem 1.2 (p. 3): the same saving for the well-factorable upper bound linear sieve weights of level D <= x^{7/12-epsilon}.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.