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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
##### Keywords:
mean value formula; asymptotic result; Hardy sum; Dedekind sum
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