Papers › The Probabilistic Zeta Function of a Finite Lattice
The Probabilistic Zeta Function of a Finite Lattice
Besfort Shala
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We study Brown's definition of the probabilistic zeta function of a finite lattice as a generalization of that of a finite group. We propose a natural alternative or extension that may be better suited for non-atomistic lattices. The probabilistic zeta function admits a general Dirichlet series expression, which unlike for groups, need not be ordinary. We compute the function for several examples of finite lattices, establishing a connection with the Stirling numbers of the second kind in the case of the divisibility lattice. Furthermore, in the context of moving from groups to lattices, we are interested in lattices with probabilistic zeta function given by ordinary Dirichlet series. In this regard, we focus on partition lattices and d-divisible partition lattices. Using the prime number theorem, we show that the probabilistic zeta functions of the latter typically fail to be ordinary Dirichlet series.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections