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Wikipedia gives the definition of the Golomb-Dickman constant as $$\lambda=\lim_{n\rightarrow\infty}\frac1n\sum_{k=2}^n\frac{\log(P_1(k))}{\log(k)}$$Where $P_1(x)$ is the largest prime factor of $x$.

My question is if $$\lim_{k\rightarrow\infty}\frac{\log(P_1(k))}{\log(k)}$$converges or not, and if so to what number? The upper and lower bounds of $P_1(k)$ are $k$ and $2$ respectively. Taking the average of the two, we get $\frac{k+2}2$. Substituting this for $P_1(k)$, we get $1$. This is a heuristic argument, so I am not sure.

Edit: I am now suspecting that the limit in question doesn't exist because of the unpredictable nature of $P_1(k)$.

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    $\begingroup$ $\log(P_1(k)) / \log(k)$ does not converge to $1$. For the subsequence $k=2^j$, the terms approach zero. $\endgroup$
    – aschepler
    Commented Sep 15, 2023 at 23:15
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    $\begingroup$ The limit does not exist: the subsequence of powers of two gives a limit of zero whereas the subsequence of primes gives a limit of one. $\endgroup$
    – TheSimpliFire ♦
    Commented Sep 15, 2023 at 23:21
  • $\begingroup$ @TheSimpliFire Is this a known theorem? What is it called? Please make this an answer - it is short but it does the job. $\endgroup$
    – Kamal Saleh
    Commented Sep 17, 2023 at 21:10
  • $\begingroup$ @KamalSaleh It is trivial. If $k$ is a prime then $P_1(k)=k$. If $k$ is a power of two then $P_1(k) =2$. $\endgroup$
    – Gary
    Commented Dec 25, 2023 at 22:59
  • $\begingroup$ @Gary I deleted this question long ago, and I undeleted. I forgot to undelete the answer as well. $\endgroup$
    – Kamal Saleh
    Commented Dec 26, 2023 at 2:28

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Credit to @TheSimpliFire:

The limit in question doesn't exist. This is because the limit is different for different subsequences (for example the primes and the powers of 2).

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