Solve $\int_1^n \exp(-s i^{-p}) i^{-p} \mathrm di=\zeta(p)\epsilon$

Question

Let $p>1,\epsilon>0, n>1$ and $s$ be the solution of the following equation

$$\int_1^n \exp\left(-s i^{-p}\right) i^{-p} \mathrm di=\zeta(p)\epsilon$$

How does $s$ depend on $n$ and $p$? For $p>1$, $s$ seems to grow sublinearly with $n$. Below are graphs of $s$ as a function of $n$ for $p=1.5$ and $p=1.8$.

notebook

1. What is the limit as $n\to \infty$?
2. Is there a nice approximation for finite $n$?

Motivation This is the integral approximation of the sum in this question, which gives a way to solve for average number of steps needed for gradient descent with learning rate 1 to decrease the objective by a factor of $1/\epsilon$ times when minimizing a quadratic with eigenvalues $\{1^{-p},2^{-p},...\}$