Brun's Inequality for a Geometric Lattice

Abstract

V. Brun introduced Brun's sieve in his seminal paper, which is based on Brun's inequality for the Möbius function and is a very powerful tool in modern number theory. The importance of the Möbius function in enumeration problems led G.-C. Rota to introduce the concept of the Möbius function for partially ordered sets. In this article, we prove Brun's inequality for geometric lattices and develop a combinatorial sieve in this context. One of the main ingredients is a recent work of K. Adiprasito, J. Huh, and E. Katz on the log-concavity of absolute values of the Whitney numbers associated with matroids. Further, we study shifted convolutions of the Whitney numbers associated with Dowling lattices and derive an asymptotic formula for generalized Dowling numbers.