A note on variants of Buchstab's identity

TL;DR

The paper broadens the toolkit for Buchstab-type sieve inequalities by introducing a large family of upper- and lower-bound iterations for sieve functions in dimensions $\kappa>1$, built from binomial-type combinatorial identities. It systematically replaces Brady's simple rules with multi-parameter variants using sets $\mathcal{U}$ and symmetric sums $M^{r}_{k}$, $N^{r}_{k}$ to derive integral inequalities linking $s^{\kappa}F_{\kappa}(s)$ and $s^{\kappa}f_{\kappa}(s)$. These results generalize existing frameworks (Selberg, DHR, Brady) and yield refined bounds for the extremal sieve functions in higher dimensions, with potential impact on estimating sifting limits $\beta_{\kappa}$. The note also situates these developments within Brady's broader set of iteration rules, inviting further generalization and applications to high-dimensional sieves.

Abstract

The author proves variants of Buchstab's identity on sieve functions, refining the previous work on new iteration rules of Brady. The main tool used in the proof is a special form of combinatorial identities related to the binomial coefficients. As a by--product, the author obtains better inequalities of $F_κ(s)$ and $f_κ(s)$ for dimensions $κ> 1$.

Theorems & Definitions (17)

• Theorem 2.1
• proof
• Corollary 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7
• Corollary 2.8