Aspects of number theory in terms of potential theory (Q1841542)

The authors are developing an interesting potential theory in the set \( \mathbb{N}^\ast\) of natural numbers strictly bigger than one. The order relation in \(\mathbb{N}^\ast\) is given by \(x\) is smaller than \(y\), if \(x\) divides \(y\). The kernel is defined by \(Vf(x)=\sum f(x/p)\), where the sum is over the primes \(p\) dividing \(x\), and its dual \(V^\ast f(x)=\sum_{p\text{\;prime}}f(px)\). The sets of extremal and subtractible elements of \(V\)-supermedian and \(V^\ast\)-supermedian functions are studied. Also Martin boundary is investigated. The set of subtractible elements induces an \(H\)-cone of functions defined in [Order and convexity in potential theory: \(H\)-cones, Lecture Notes in Mathematics 853, Springer Verlag (1981; Zbl 0534.31001)] by \textit{N. Boboc, Gh. Bucur} and \textit{A. Cornea}. Some results of the earlier paper [Rev. Roum. Math. Pures Appl. 40, 259-277 (1995; Zbl 0856.31008)] by the authors are improved.