2025/09/29 by Hartosh Singh Bal, Bal, Hartosh Singh
Computer Science · Mathematics · #11A07 #11R18 #13F35 #37C25 #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #History and Theory of Mathematics #Mathematical and Theoretical Analysis #Number Theory (math.NT) #Primary 11B37 #Secondary 05A17
paper · pdf · doi:10.48550/arxiv.2509.25038
openalex publication_date 2025/09/29 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
We develop a Witt--Hadamard calculus for Euler products that unifies the classical Gauss congruences with their modern refinement, the Dold congruences. Within this framework we prove norm descent: Dold congruences are functorial under finite extensions and preserved by prime--ideal norms NK/ℚ, yielding integer ghosts from algebraic ones. We extend the theory from ℤ to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Two rigidity theorems lie at the core: a cyclotomic residues theorem, asserting that if the logarithmic derivative has only cyclotomic poles then integrality forces rationality; and a stronger \emphDold+ rigidity theorem, showing that any algebraic series satisfying refined Dold congruences is necessarily rational. These results sharpen the Gauss--Dold picture: ordinary congruences enforce integrality, while the strengthened form collapses algebraic cases to rational ones. Applications include prime--ideal ladders in number fields and exact product laws for dynamical zeta functions, illustrated for subshifts of finite type and circle doubling.