CIIM 2021 Problem 5

For every positive integer $n$ , let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$ ; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$ . If we define $f(n)=(-1)^{s(n)}$ , prove that the limit $$\lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n}$$ exists and determine its value.