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A sum analogous to the Dedekind sum and its mean value formula. (English) Zbl 0997.11076

Let ((x))=x-[x]-1 2 if x is not an integer, and let ((x))=0 otherwise. The sum in the title, originally introduced by Hardy, is defined by

S 2 (h,k)= a=1 k (-1) a a kah k·

For even k it can be expressed in terms of the classical Dedekind sum S(h,k) by the relation

S 2 (h,k)=2S(h,k/2)-S(h,k)·

The author uses some of his earlier work on S(h,k) to deduce corresponding results for S 2 (h,k). A corollary of the main theorem gives, for an odd prime p,

' h=1 2p |S 2 (h,2p)| 2 =1 32p 2 +Op exp 4logp loglogp·

11N37Asymptotic results on arithmetic functions
11F20Dedekind eta function, Dedekind sums