0
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ What's the question? $\endgroup$ – Wojowu May 4 at 14:42
  • $\begingroup$ @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). $\endgroup$ – Ofir Gorodetsky May 4 at 16:27
0
$\begingroup$

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.

As is often the case in this kind of problems, this was answered by Tenenbaum and by Tenenbaum-Wu, who improve on earlier works by Song and Greaves.

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 $x<y$, 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 $F(x,y)/\log y \asymp \Psi(x,y)/(x\log x)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.