Suppose that $f \colon [1,\infty) \to \mathbb{Z}^{+}$ is a function such that the formula
$$f(x) = \frac{\pi(x)}{\sqrt{\frac{\pi}{2}\log_{2}x}} \left(e^{-\frac{(k-0.5\log_{2} x)^{2}}{0.5 \log_{2}x}}+\mathcal{O} ((\log x)^{-0.5 +\varepsilon})\right)$$
is said to hold uniformly for all integers $k\geq 0$ and any given $\varepsilon>0$. (As usual, $\pi(x)$ is the number of primes in the interval $[1,x]$.)
Is it possible to deduce from such a formula that if $k$ is sufficiently large then $f(x)\geq 1$ for infinitely many $x$'s? In the affirmative case, how easy was it for you to see what the answer was?
Thanks in advance for taking the time to leave your observations or answers.