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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 $\sum_{n \leq x}1/\varphi(n)$?

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  • $\begingroup$ It's Tenenbaum (Gérald) who by the way is not only a number theorist but also a writer. $\endgroup$ – Sylvain JULIEN Jun 15 '17 at 19:03
  • $\begingroup$ For Question 1 - do not both A and also the well known asymptotic expansion$$H_n\sim \ln{n}+\gamma+\sum_{k=1}^\infty \frac{B_k}{k n^{k}}=\ln{n}+\gamma+\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4}-\cdots$$imply B easily? $\endgroup$ – მამუკა ჯიბლაძე Jun 15 '17 at 20:15
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    $\begingroup$ @მამუკაჯიბლაძე: It surely does, but there is the subtlety that the OP's first display is valid for integer $x\geq 1$, while the second display is valid for all real $x\geq 1$. $\endgroup$ – GH from MO Jun 15 '17 at 20:43
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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.

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