Some identities involving the Riemann zeta function
The paper presents six identities of which three are due or partially due to other authors. They are grouped in pairs of identities which are equivalent by Möbius inversion. As an example, if \(a\) is an integer \(\ge 2\), then \[ 2 \sum_{r=1}^\infty \frac1{r^a} \sum_{k=1}^r \frac1k = (a+2)\zeta(a+1) - \sum_{i=1}^{a-2} \zeta(a - i) \zeta(i+i) \] is identity (1.1). Dividing through by \(\zeta(a+1)\) yields (1.1a) with left side \[ 2 \sum_{r=1}^\infty \frac1{r^n} \sum_{\substack{k=1 \\ (k,r)=1}}^r \frac1k \] after taking the Dirichlet product. The other identities are vaguely similar, but without the parameter \(a\). By taking \(a = 2\) and dividing by 2 in either of the above and subtracting the corresponding member of the second pair, they get the third. Thus, it suffices to prove (1.1) and (1.2) to prove them all. This is accomplished by means of a generalization of a transformation formula due to \textit{J. Lehner} and \textit{M. Newman} [Acta Arith. 15, 181--187 (1969; Zbl 0176.32304)] and a reciprocity relation involving \(\zeta\). Other authors cited are \textit{T. M. Apostol} [Modular functions and Dirichlet series in number theory. New York etc.: Springer (1976; Zbl 0332.10017), p. 111], Williams (see Klamkin, Solution of problem 4564, Am. Math. Mon. 62, p. 130), \textit{W. E. Briggs}, \textit{S. D. Chowla}, \textit{A. J. Kempner} and \textit{W. E. Mientka} [Scripta Math. 21, 28--30 (1955; Zbl 0065.30401)], \textit{R. J. Hans} and \textit{V. C. Dumir} [Res. Bull. Panjab Univ., n. Ser. 15, 353--356 (1965; Zbl 0135.10402)] and \textit{H. Gupta} [Res. Bull. Panjab Univ., n. Ser. 15, 347--349 (1965; Zbl 0135.10401)].
scientific article; zbMATH DE number 3619967