summation of this series as $ x \to \infty $ ??

Question

given the series for the Mangoldt function $ \Lambda (n) $

$$ f(x)= \sum_{n=1}^{\infty}\frac{\Lambda (n)}{\sqrt{n}}\cos(\sqrt{x} \log n+\pi /4) $$

if we truncate the series, can we say that

$$\lim_{x\to\infty} \frac{f(x)}{x^{1/4}}=0$$

Answer 1

Hint:

$$ \lim_{x\to \infty} \frac{\cos( a\sqrt{x}+b )}{x^{1/4}}=0, $$

since

$$ \Big|\frac{\cos( a\sqrt{x}+b )}{x^{1/4}}\Big| \leq \frac{2}{x^{1/4}}. $$