# A sum with integer parts

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.

• $k\log(k+1)$, an integer?? Jul 28 '17 at 14:18
• 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 $\sum_{k \in A,k \le x}1 \sim \int_0^x f(y)dy$ Jul 28 '17 at 15:10
• @ Jean Duchon : Oops, I just corrected. Jul 28 '17 at 16:00
• One could also take the integer part of $k \log(k + 1)$ or take the prime numbers. Jul 28 '17 at 16:11
• @ 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 $\pi_{ \mathcal{A} } = \sum_{a \in \mathcal{A} } \delta_a$. Jul 28 '17 at 16:44