{"id":"palomar/PALOMAR-2026-10-20-000005","url":"https://github.com/jkfarland-bot/DedekindSubgroups-lean/tree/cb87faf157821154dc1beef7aad753d889211ad9","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-20-000005-v1.json","repo":"https://github.com/jkfarland-bot/DedekindSubgroups-lean","tags":["math.NT","msc:11F20","msc:11M20","msc:68V20","dedekind-sums","number-theory","congruences","lean-formalization"],"score":0,"title":"DedekindSubgroups: a congruence modulo 16 for sums of Dedekind sums over subgroups of odd order","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"For k >= 2 and a subgroup H of odd order n of the unit group (Z/kZ)^x, the sum S(H,k) of the Dedekind sums s(h,k) over h in H satisfies: 12k S(H,k) is an integer congruent to n(k-1)(k-2) modulo 16, and the modulus 16 cannot be replaced by 32 (an explicit pair of order-3 subgroups of (Z/63Z)^x). The development also proves a congruence modulo 8 for every subgroup, a congruence for 2k S(H,k) when 3 does not divide k, and, for primes p > 3 and n > 1, that 2S(H,p) is an integer congruent to n(p-1)/2 modulo 8, which refines Louboutin's parity theorem and gives 12S(H,p) - p congruent to 3n(p-1) -…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-20","reactions":{},"formalized":true,"formalizes":[{"title":"A congruence modulo 16 for sums of Dedekind sums over subgroups of odd order","identifier":"https://doi.org/10.5281/zenodo.23425872","relationship":"formalizes"}]}