2017/04/21 by Ken Ono, Robert Schneider, Ono, Ken +3
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1704.06636
openalex publication_date 2017/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If gcd(r,t)=1, then a theorem of Alladi offers the Möbius sum identity -∑_\substack n ≥ 2
p_\rmmin(n) ≡ r \pmodt μ(n)n-1= (1)/(φ(t)). Here p_\rmmin(n) is the smallest prime divisor of n. The right-hand side represents the proportion of primes in a fixed arithmetic progression modulo t. Locus generalized this to Chebotarev densities for Galois extensions. Answering a question of Alladi, we obtain analogs of these results to arithmetic densities of subsets of positive integers using q-series and integer partitions. For suitable subsets § of the positive integers with density d_§, we prove that - limq → 1 ∑_\substack λ∈ P
\rmsm(λ) ∈ § μP (λ)q\vert λ\vert = d_§, where the sum is taken over integer partitions λ, μP(λ) is a partition-theoretic Möbius function, \vert λ\vert is the size of partition λ, and \rmsm(λ) is the smallest part of λ. In particular, we obtain partition-theoretic formulas for even powers of π when considering power-free integers.