Question 29
Prove that for every positive integer $k$ there exists an integer $n$ and distinct primes $p_1, p_2, \dots, p_k$ such that, if $A(n)$ denotes the number of integers in $\{1, 2, \dots, n\}$ which are relatively prime to $p_1p_2\dots p_k$, then
$$
\left| n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \cdots \left(1 - \frac{1}{p_k}\right) - A(n) \right| \gt 2^{k-3}.
$$
$$
\left| n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \cdots \left(1 - \frac{1}{p_k}\right) - A(n) \right| \gt 2^{k-3}.
$$