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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 $p$, $$\mathop{{\sum}'}_{h=1}^{2p} |S_2(h,2p)|^2= \frac{1}{32} p^2+ O\Biggl(p\exp \biggl( \frac{4\log p}{\log\log p} \biggr)\Biggr).$$

11N37Asymptotic results on arithmetic functions
11F20Dedekind eta function, Dedekind sums
Full Text: DOI
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