# Does $\lim_{k\rightarrow\infty}\frac{\log(P_1(k))}{\log(k)}$ exist?

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 $$P_1(k)$$.

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