Weighted Riemann Zeta Function Limits on the Real Axis

In the paper, it is investigated whether for real argument $$s$$ the $$(s-1)^{n+1}$$ weighted Riemann zeta function $$\zeta^{(n)}(s)$$ limits $$s\downarrow 1$$ do exist. Here we will look only at $$n=0,1$$ . The answer to the question could very well be that assuming existence of limits to be true gives a conflicting outcome. This result may support the possibility of what can be called incompleteness in concrete mathematics. Interestingly, the uncovered incompleteness occurs in an approximation-theoretical framework. The key argument we employ in the paper revolves around the approximation of a l’Hôpital limit. Only an ‘‘after the fact’’ alternative approach reorders the formulae in the derivation. This reordering is, in fact, based on a $$0=1$$ contradiction as starting point. However, the alternative $$0=1$$ based approach is not the limit approximation that is used in the flow of the derivation in the present paper.