The error term in a result of Landau 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 $\sum_{n \leq x}1/\varphi(n)$?
 A: Please exercise more care when quoting a result. The first result is Theorem 0.5 from Tenenbaum: Introduction to analytic and probabilistic number theory. The correct form of this result is
$$\sum_{n \leq x}\frac{1}{n}=\log x +\gamma+\frac{1}{2x}-\frac{1}{12x^2}+\frac{\alpha}{\color{red}{60}x^4},$$
where $\alpha\in(0,1)$, and it is valid for positive integers $x$ only.
The second result holds (for arbitrary real $x\geq 1$) in the form
$$\sum_{n \leq x}\frac{1}{n}=\log x +\gamma\color{red}-\frac{\{x\}-1/2}{x}+O\left (\frac{1}{x^2} \right ),$$
and it can be proved easily with Euler-Maclaurin summation (specifically by two integration by parts).
The quoted article (which is available freely here) mentions an earlier paper by the same author (Indian J. Pure Appl. Math. 13 (1982), 882-885) which improves Landau's error term $O(\log x/x)$ to $O((\log x)^{2/3}/x)$. Based on the MathSciNet quotations of these two papers, in particular a 2014 paper by Sankaranarayanan and Singh, this appears to be the best known error term at the moment.
