Some number theoretic results (Q2546057)

The paper consists of two disconnected parts. In the first part the authors investigate the maximal number \(f(p)\) of residues \(\mod p\) so that sums of different number of distinct elements of our set are distinct \(\mod p\). They prove \[ (4p)^{1/3}+o(p^{1/3})<f(p)<(288p)^{1/3}+o(p^{1/3}). \] It is conjectured that the lower bound gives the correct order of magnitude. In the second part they investigate the irrationality of series of the form \[ \sum ^\infty_{n=1}{u_n \over a_1,a_2 \ldots a_n},a_1 \leq a_2 \leq \ldots . \]