# Sum of reciprocals of friable (i.e. smooth) numbers

Consider the `logarithmic' sum $$F(x,y) = \sum_{n > x, \, n \text{ is }y\text{-friable}} \frac{1}{n}.$$

What are it's asymptotics (for general $$x,y\ge 1$$).

I would expect that this sum was studied in the multiplicative number theory literature and would appreciate a reference for its asymptotics.

I know how I would go about this myself (study the corresponding Perron integral via saddle point analysis or other approaches, depending on the parameters). It is also not hard to work out $$F$$'s order of magnitude, using Hildebrand and Tenenbaum's results on $$\Psi(cx,y)/\Psi(x,y)$$ where $$\Psi(x,y) = \#\{n \le x: n \text{ is } y \text{-friable}\}$$.

EDIT: In the answer below I describe how a much more general problem was studied extensively. However, results for small $$y$$ are lacking.

• What's the question? – Wojowu May 4 at 14:42
• @Wojowu What is the asymptotics of this sum, in all ranges of $x$ and $y$ (a proof will do, of course, but I expect this was studied already). – Ofir Gorodetsky May 4 at 16:27

After enough searches, I've found the answer in the literature, at least for $$y$$ which is not too small w.r.t $$x$$. I would still appreciate input on smaller $$y$$, or on older references.
The general problem that has been studied is the behaviour of $$\sum_{n \le x, n \text{ is }y\text{-friable}} f(n)/n$$ when $$x \ge y \ge 2$$, for multiplicative functions $$f$$ with $$\sum_{p \le t} f(p) \log p$$ close to $$\kappa \log t$$ for some $$\kappa >0$$. Given precise enough results, one can recover asymptotics for $$\sum_{n > x, n \text{ is }y\text{-friable}} f(n)/n$$, noticing that
$$\sum_{n \le x, n \text{ is }y\text{-friable}} f(n)/n + \sum_{n > x, n \text{ is }y\text{-friable}} f(n)/n = V_{f}(y), \qquad V_f(y) := \prod_{p \le y} \sum_{k \ge 0} f(p^k)/p^k.$$
I am only interested in $$f \equiv \mathbf{1}$$ (so $$\kappa = 1$$). If $$x, evidently $$F(x,y) = V_{\mathbf{1}}(y) - \sum_{n \le x} 1/n \sim e^{\gamma} \log y \left( 1 - \frac{\log x}{e^{\gamma}\log y}\right)$$ by Mertens' third theorem. If $$x \ge y$$, we appeal to the main Theorem in Tenenbaum, ''Note on a paper by J. M. Song'', Acta Arith. 97 (2001), no. 4, 353–360, and find $$F(x,y) = V_{\mathbf{1}}(y) \left(1- j_1( \log x/ \log y)(1+O(\log^{-\delta} y)) \right)$$ for every $$\delta \in (0,1)$$, where $$j_1(u)$$ is defined in terms of a delayed differential equation. In p. 4 of Tenenbaum and Wu, ''Moyennes de certaines fonctions multiplicatives sur les entiers friables. III'', Compos. Math. 144 (2008), no. 2, 339–376, our sum is denoted by $$\psi^{*}_{\mathbf{1}}(x,y)$$ and it is explained that $$1- j_1(u) = \lambda_1(u), \qquad \lambda_1(u) := e^{-\gamma}\int_{u}^{\infty} \rho(v) \, d v$$ where $$\rho$$ is the Dickman function, and one has the large-$$u$$ asymptotics $$\lambda_1(u) \sim \frac{e^{-\gamma}\rho(u)}{\log u}.$$ The error term $$O(\log^{-\delta}y)$$ in Tenenbaum can overtake the main term if $$y$$ is small w.r.t $$x$$; Theorem 3.2 in the aforementioned Tenenbaum-Wu paper improves the range.
To summarize, $$\frac{F(x,y)}{\log y} \asymp \frac{\rho(u)}{\log (u+1)}$$ as $$x \to \infty$$ if $$y\ge 2$$ is not too small w.r.t to $$x$$, where $$u = \max\{ \log x/ \log y, 1\}$$. For $$u \to \infty$$ this is an asymptotic result, while for bounded $$u$$ the constant is different.
This still leaves open the behaviour for small $$y$$. E.g. for $$y=2$$, $$F(x,y)/\log y \asymp 1/x$$ while $$\rho(u)/\log(u+1)$$ is much smaller. A naive guess is that the order of magnitude in this case is $$F(x,y)/\log y \asymp \Psi(x,y)/(x\log x)$$.