DedekindSubgroups: a congruence modulo 16 for sums of Dedekind sums over subgroups of odd order
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) -…
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