Math. Proc. Camb. Phil. Soc. (1980), 88, 11 Printed in Great Britain Dedekind sums and Hecke operators BY L. ALAYNE PARSON Ohio State University, Columbus, Ohio 43210 (Received 10 September 1979) Introduction. By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the trans- formation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.) 1. The classical Dedekind sumand the Petersson- Knopp identity. Ifh,kelwithk > 0, the classical Dedekind sum s(h, k) is defined by 0 if xll. This sum has been studied extensively. A detailed account of many of its properties may be found in (10) and (11). Two simple properties which we shall need later are ) = s(h,k), (1-1) s(nh,nk) = s(h,k) for nel+. (1-2) (1-1) follows immediately from the definition of s(h, k) whereas (1.2) is derived in (ll). Dedekind's work(8) contains the identity P-I s(ph,k)+ 2 s(h + mk,pk) = (p+l)s(h,k) (1-3) m = 0 for p prime. An elementary proof of (1-3) may be found in Rademacher and White- man(ii). M.Knopp's(8) extension of this identity is 2 2 s(ah + bk,dk) = a(n)s(h,k), (1-4) ad=n b (mod d) d>0