Let $x$ and $y$ be two positive real numbers. What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum $$\sum_{\substack{n \leq x \\ P(n)\leq y}} \log(n),$$ where $P(n)$ is the greatest prime factor of $n.$ Thanks in advance.
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2$\begingroup$ It should be close to $K(y)x\log(x/e)$ with $K(y)$ a version of Dickman's constant. (Maybe $K(\log y/\log x)$ is more standard.) You might get some improvement by a clever pairing of smooth numbers, say log(m)d(m)/2 for m a certain smooth number m larger than x. This might get you most of the sum, and then you can approximate the rest with a small multiple of log x. Gerhard "Or Try Two Large Smooths" Paseman, 2018.08.09. $\endgroup$– Gerhard PasemanAug 9, 2018 at 10:25
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2$\begingroup$ I'd never heard the term 'friable' for integers before. Is 'smooth' used interchangeably with 'friable'? (Perhaps it's a matter of linguistic background, since the paper you cite below is in French?) $\endgroup$– LSpiceAug 9, 2018 at 13:24
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3$\begingroup$ @LSpice: Some people prefer friable, as the word smooth already carries too many different meanings. For others friable sounds too much like fryable. $\endgroup$– Jan-Christoph Schlage-PuchtaAug 9, 2018 at 15:08
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1$\begingroup$ @Jan-ChristophSchlage-Puchta, well, there’s no denying that ‘friable’ is a much mathematically rarer term (as well as being quite apposite). In my field, we have the very inventive term ‘good’. :-) $\endgroup$– LSpiceAug 9, 2018 at 18:18
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De La Breteche and Tenenbaum established in their paper 'Propriétés statistiques des entiers friables' the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 \leq y \leq x,$ we have $$\sum_{\substack{n \leq x \\ P(n)\leq y}} \log(n)=\left\{\log(x)-\frac{\log(y)+O(\log\log(y))}{\log\left(1+\frac{y}{\log(x)}\right)} \right\}\Psi(x,y),$$ where $\Psi(x,y)$ is the number of friables integers less than $x$ whose greatest prime factors are less than $y.$