On the Nature of gamma-th Arithmetic Zeta Functions

Abstract

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite-Lindemann theorem (about the transcendence of e^\alpha, for algebraic values of \alpha) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as e^{\gamma \log n}. This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called gamma -th arithmetic zeta function, for a positive integer n and a complex number \gamma. Moreover, we raise a conjecture about the exceptional set of \zeta_\gamma, in the case in which \gamma is transcendental, and we connect it to the famous Schanuel's conjecture.