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A sum analogous to the Dedekind sum and its mean value formula. (English) Zbl 0997.11076
Let \(((x))= x-[x]- \frac{1}{2}\) if \(x\) is not an integer, and let \(((x))=0\) otherwise. The sum in the title, originally introduced by Hardy, is defined by \[ S_2(h,k)= \sum_{a=1}^k (-1)^a \biggl(\biggl( \frac{a}{k} \biggr)\biggr) \biggl(\biggl( \frac{ah}{k} \biggr)\biggr). \] 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 \(p\), \[ \mathop{{\sum}'}_{h=1}^{2p} |S_2(h,2p)|^2= \frac{1}{32} p^2+ O\Biggl(p\exp \biggl( \frac{4\log p}{\log\log p} \biggr)\Biggr). \]

MSC:
11N37 Asymptotic results on arithmetic functions
11F20 Dedekind eta function, Dedekind sums
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