Equality of Dedekind sums mod ℤ, 2ℤ and 4ℤ

In [K. Girstmair, A criterion for the equality of Dedekind sums mod ℤ, Int. J. Number Theory10 (2014) 565–568], it was shown that the necessary condition b ∣ (a₁a₂ - 1) × (a₁ - a₂) for equality of two Dedekind sums s(a₁, b) and s(a₂, b) given in [S. Jabuka, S. Robins and X. Wang, When are two Dedekind sums equal? Int. J. Number Theory7 (2011) 2197–2202] is equivalent to 12s(a₁, b) - 12s(a₂, b) ∈ ℤ. In this paper, we give a new proof of this result and then find two additional necessary and sufficient conditions for 12s(a₁, b) - 12s(a₂, b) ∈ 2ℤ, 4ℤ. These give new necessary conditions on equality of Dedekind sums.