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Based on an equivalence relation that was established recently on exponentialsums, in this paper we study the class of functions that are equivalent to theRiemann zeta function in the half-strip$\{s\in\mathbb{C}:\operatorname{Re}s>1\}$. In connection with this class offunctions, we first determine the value of the maximum abscissa from which theimages of any function in it cannot take a prefixed argument. The main resultshows that each of these functions experiments a vortex-like behavior in thesense that the main argument of its images varies indefinitely near thevertical line $\operatorname{Re}s=1$. In particular, regarding the Riemann zetafunction $\zeta(s)$, for every $\sigma_0>1$ we can assure the existence of arelatively dense set of real numbers $\{t_m\}_{m\geq 1}$ such that theparametrized curve traced by the points$(\operatorname{Re}(\zeta(\sigma+it_m)),\operatorname{Im}(\zeta(\sigma+it_m)))$,with $\sigma\in(1,\sigma_0)$, makes a prefixed finite number of turns aroundthe origin.

The vortex-like behavior of the Riemann zeta function to the right of the critical strip