Abstract: Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of e α , for algebraic values of α ) 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 γ 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 γ-th arithmetic zeta function ζ γ ( n ) : = n γ ( = e γ log n ), for a positive integer n and a complex number γ . Moreover, we raise a conjecture about the exceptional set of ζ γ , in the case in which γ is transcendental, and we connect it to the famous Schanuel’s conjecture.

Abstract: Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for $\zeta(3)$ due to Markov and rediscovered by Ap\'ery. In this paper we extend Koecher's method to a very general setting and prove two more specific but still rather general results. As applications we obtain infinite classes of identities for alternating Euler sums, further Markov-Ap\'ery type identities, and identities for even powers of $\pi$

Abstract: The zeta-functions associated with algebraic curves over finite fields encode many arithmetic properties of the curves. In the non-singular case the theory is well-known. It is analogous to the theory of zeta-functions for number fields and culminates in the Hasse-Weil theorem about the Riemann hypothesis for curves. In the singular case, the theory is more difficult and less explored.First of all, one does not deal with Dedekind rings anymore, but with orders, i.e., certain subrings of them. The corresponding theory of(fractional) ideals becomes much more complicated. Secondly, there are various candidates for the zeta-function. We follow here the approach of K. O. Stöhr. His zeta function is defined in terms of positive ideals, and it has a functional equation and an Euler product. It differs from the zeta function of the associated smooth curve (the mormalization) only in the finitely many factors at the singular points. Therefore it suffices to study the zeta functions of local (singular) orders, which is the topic of this thesis. After discussing some general results we concentrate on the rational unibranch case, i.e., the case where the associated maximal order is local again, with residue field equal to the ground field. Extending results of Stöhr, we provide tools to calculate these zeta-functions in an effective way. Using this we determine the zeta functions for all orders of singularity degree at most 3, and for a series of orders which we cal balanced. Then we study when the Riemann hypothesis hold for the considered orders. Stöhr has given examples where it fails. We show that the Riemann hypothesis holds for al balanced orders, but give some evidence that it fails in all other cases.