The standard asymptotic for the harmonic sum has a form such as

A) $$\sum_{n \leq x}\frac{1}{n}=\log x +\gamma+\frac{1}{2x}-\frac{1}{12x^2}+\frac{\alpha}{65x^4},$$ where $\alpha \in (0,1)$, see Probabilistic Number Theory, Tanenbaum, Theorem 0.8.

There are two published results based on the formula

B) $$\sum_{n \leq x}\frac{1}{n}=\log x +\gamma+\frac{\{x\}-1/2}{x}+O\left (\frac{1}{x^2} \right ),$$ where $\{x\}=x-[x]$ is the fractional part, see "On an error term of Landau II", Sidaramachandrarao, Rocky Mount. Math journal, Vol. 15, No. 2, 1985, page 583, equation (2.9).

Question 1. Is formula B correct?

Question 2. Does Landau result remains as the best error term for the sum $\sum_{n \leq x}1/\varphi(n)$?