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Source. Definition 1.1, p. 2, of Alexandru Pascadi, On the exponents of distribution of primes and smooth numbers, arXiv:2505.00653v2 (29 June 2025), the version named on the source card. A preprint. The paper recalls the definition from Maynard's work on primes in arithmetic progressions to large moduli.

Read depth. Claims checked: the definition was read clause by clause on the page image. Nothing here is independently reviewed.

Statement

Definition 1.1 (p. 2). A complex sequence $(\lambda_q)_{q\le Q}$ is triply-well-factorable of level $Q$ when, for every choice of $Q_1,Q_2,Q_3\ge1$ with $Q_1Q_2Q_3=Q$, there are 1-bounded complex sequences $(\alpha_{q_1})$, $(\beta_{q_2})$, $(\gamma_{q_3})$ supported on $q_i\le Q_i$ such that for every $q$

$$\lambda_q=\sum_{q_1q_2q_3=q}\alpha_{q_1}\beta_{q_2}\gamma_{q_3}.$$

The paper notes (p. 2) that such weights arise in a slight variant of the $\beta$-sieve with $\beta\ge2$, and (p. 24) that Iwaniec's well-factorable linear sieve weights factor in two pieces at every split but are not triply-well-factorable in this sense.

Dependencies

None.

Bears on

No Erdős problem page in the corpus links this definition. It is the weight class of Theorem 1.3 (i).