Hecke operators and an identity for the dedekind sums

Abstract

If h, k ∈ Z, k > 0, the Dedekind sum is given by s(h,k) = ∑ μ=1 k μ k hμ k , with ((x)) = x − [x] − 1 2 , x∉Z , =0 , x∈Z . The Hecke operators T n 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.