Primitivity index bounds in free groups, and the second Chebyshev function

Motivated by results about “untangling” closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, $d_{prim}(g;F_N)$ and $d_{simp}(g;F_N)$, where $1\mathrm{\ne }g\in F_N$, and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence $d_{prim}(a^n{b}^n;F(a,b))$ as $n\to \mathrm{\infty }$. Answering a question of Kapovich, we prove that this sequence is unbounded and that for $n_i=lcm(1,2,\dots ,i)$, we have $\mathrm{\mid }{d}_{prim}(a^{n_i}{b}^{n_i};F(a,b))-\log (n_i)\mathrm{\mid }\le o(\log (n_i))$. By contrast, we show that for all $n\ge 2$, one has $d_{simp}(a^n{b}^n;F(a,b))=2$. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.