On the generating functions of Mersenne and Fermat primes (Q692048)

Let \(\mathbf{M}\) and \(\mathbf{F}\) denote the sets of Mersenne and Fermat primes, respectively. Assume that \(\nu(d)\) is an arbitrary completely additive function with \(\nu(d)\ll d^k\) for some \(k>0\). In the paper under review, the author applies a generalization of the so called Golomb's formula to obtain expressions including the summations \(\sum_{p\in\mathbf{M}}\nu(p)z^{2p+1}\) and \(\sum_{p\in\mathbf{F}}\nu(p)z^{2p-1}\). As an application, the author formulates a limit relation, in which under assumption that it is not true, then the set \(\mathbf{M}\cup\mathbf{F}\) is infinite.