Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing

In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Mu\~noz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture by Weng for the ranks 2 and 3, and we prove the counting miracle conjecture.