Hecke operators and an identity for the dedekind sums

AbstractIf h, k ∈ Z, k > 0, the Dedekind sum is given by s(h,k) = ∑μ=1kμkhμk, with ((x)) = x − [x] − 12, x∉Z, =0 , x∈Z. The Hecke operators Tn for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z+) ∑ ∑ s(ah+bk,dk) = σ(n)s(h,k), ad=n b(mod d)d>0 where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (∗) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (∗) when n is prime