Limit heights and special values of the Riemann zeta function

We study the distribution of the height of the intersection between the projective line defined by the linear polynomial $x_{0}+x_{1}+x_{2}$ and its translate by a torsion point. We show that for a strict sequence of torsion points, the corresponding heights converge to a real number that is a rational multiple of a quotient of special values of the Riemann zeta function. We also determine the range of these heights, characterize the extremal cases, and study their limit for sequences of torsion points that are strict in proper algebraic subgroups. In addition, we interpret our main result from the viewpoint of Arakelov geometry, showing that for a strict sequence of torsion points the limit of the corresponding heights coincides with an Arakelov height of the cycle of the projective plane over the integers defined by the same linear polynomial. This is a particular case of a conjectural asymptotic version of the arithmetic B\'ezout theorem. Using the interplay between arithmetic and convex objects from the Arakelov geometry of toric varieties, we show that this Arakelov height can be expressed as the mean of a piecewise linear function on the amoeba of the projective line, which in turn can be computed as the aforementioned real number.