On summing to arbitrary real numbers (Q841461)

With real \(a_n \neq 0, n \geq 1\), let \((*)\) \(\sum 1/a_n\) be conditionally convergent; be \(\lambda\leq\Lambda\) whichever. Then, according to B. Riemann, there exists a rearrangement \(\sum 1/b_n\) of \((*)\) such that \(\liminf \sum^N 1/b_n = \lambda\), \(\limsup \sum^N 1/b_n =\Lambda\). The present paper says that there exists a sequence of \(c_n\in \mathbb N\) such that the partial sums of \((**)\) \(\sum^\infty 1/(a_n c_n)\) have \(\lambda,\Lambda\) as their liminf and limsup, respectively. Thus, all of \(\mathbb R\) is covered by values of the form \((**)\). (Compare to Theorem 1 in [\textit{J. Hančl}, Acta Arith. 59, No. 2, 97--104 (1991; Zbl 0701.11005)].)