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Let $ \mathcal{A} $ be a set of reals such that $ \sum_{a \in \mathcal{A} } \frac{1}{a} = \infty $ and $ \sum_{a \in \mathcal{A} } \frac{1}{a^2} < \infty $. For instance, $ \mathcal{A} = \mathbb{N}^* $ or $ \mathcal{A} = \{ k \log(k + 1), k \in \mathbb{N}^* \} $.

Let $ [x] $ denote the integer part of $x$. Define $ \varphi_n $ for $ x, y \in \mathcal{A} $ by $$ \varphi_n(x, y) := \frac{1}{n} \left[ \frac{n}{xy } \right] - \frac{1}{n} \left[ \frac{n}{x \vphantom{y} } \right] \frac{1}{n} \left[ \frac{n}{ y } \right] \quad \textrm{if} \quad x \neq y, \quad \textrm{and} \quad \varphi_n(x, x) = 0. $$

I am interested in the asymptotic behaviour of the sum $$ S_n(\mathcal{A} ) := \sum_{ x, y \in \mathcal{A} } \varphi_n(x, y) = \sum_{ x, y \in \mathcal{A}, xy \leqslant n } \varphi_n(x, y) $$

If $ \mathcal{A} = \mathbb{N}^* $, one can use Riemann sums to prove that $ S_n(\mathbb{N }^*) $ diverges logarithmically (like $ \log(n)^2 $ to be precise). But what about $ \mathcal{A} = \{ k \log(k + 1), k \geqslant 1 \} $ ? For this last set, numerics show a limit.

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    $\begingroup$ $k\log(k+1)$, an integer?? $\endgroup$ Jul 28 '17 at 14:18
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    $\begingroup$ I would estimate $\int_0^\infty \{\frac{n}{x}\} f(x)dx =n\int_0^\infty \{x\} f(\frac{n}{x})\frac{dx}{x^2}$ where $\sum_{k \in A,k \le x}1 \sim \int_0^x f(y)dy$ $\endgroup$
    – reuns
    Jul 28 '17 at 15:10
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    $\begingroup$ @ Jean Duchon : Oops, I just corrected. $\endgroup$
    – Synia
    Jul 28 '17 at 16:00
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    $\begingroup$ One could also take the integer part of $ k \log(k + 1) $ or take the prime numbers. $\endgroup$
    – Synia
    Jul 28 '17 at 16:11
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    $\begingroup$ @ reuns : indeed, one can write $$ \varphi_n(x, y) = -\frac{1}{n} \left\lbrace \frac{n}{xy} \right\rbrace - \frac{1}{n} \left\lbrace \frac{n}{x \vphantom{y} } \right\rbrace \frac{1}{n} \left\lbrace \frac{n}{ y } \right\rbrace + \frac{1}{n} \left\lbrace \frac{n}{ y } \right\rbrace \frac{1}{x} + \frac{1}{n} \left\lbrace \frac{n}{ x \vphantom{y} } \right\rbrace \frac{1}{y} $$ and the problem amounts then to study $ \int_0^{+\infty} \left\lbrace \frac{n}{ x \vphantom{y} } \right\rbrace d\pi_{ \mathcal{A} }(x) $ where $ \pi_{ \mathcal{A} } = \sum_{a \in \mathcal{A} } \delta_a $. $\endgroup$
    – Synia
    Jul 28 '17 at 16:44

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