What is known about the mode of the number of divisors $\le x$?

Let $$d(x)$$ be the divisor function. Let $$M(x)$$ ($$x$$ a positive integer) be the most frequent value of $$d(y)$$ for $$1 \le y \le x$$. Is an asymptotic known for $$M(x)$$, and failing that, can $$M(x)$$ at least be shown to diverge? [EDIT: Stanley Yao Xiao points out in the comments that $$M(x)$$ will certainly diverge by the asymptotic of Landau given below, but I am still curious about whether an asymptotic is known for $$M(x)$$.]

Below is some information about the problem.

My friend checked the first $$100,000$$ or so values of $$x$$ and found that the most common value of $$d(x)$$ was $$4$$ in that range. However, by a classical estimate of Landau [1], the number of $$k$$-almost primes (i.e., numbers with exactly $$k$$ prime factors, counted with multiplicity) less than or equal to $$x$$ is asymptotic to $$\frac{x(\log \log x)^{k-1}}{(k-1)!\log x}.$$

Therefore, I expect $$M(x)$$ to increase without bound, but I cannot find a proof even of this. Indeed, it seems difficult to find papers about the mode of any arithmetic function. I did find a 1994 result of De Koninck [2] that the mode of the largest prime divisor of integers in $$[2,x]$$ is asymptotic to $$\exp\left(\sqrt{\frac{1}{2}\log x(\log \log x + \log \log \log x + O(1))}\right).$$

(This result is printed as $$o(1)$$ in the original work, but it is corrected in [3] to $$O(1)$$.)

Moreover, there is a paper by Erdős and Mirsky [4] in which they prove that if $$D(x)$$ is the number of distinct values assumed by $$d(n)$$ for $$1 \le n \le x$$, then $$D(x)$$ is asymptotic to $$\frac{2\pi \sqrt{2}}{\sqrt{3}} \frac{(\log x)^{1/2}}{\log \log x}.$$

However, I could not find information about the mode of $$d(x)$$ from a quick look at that paper. This does not rule out the possibility of it being buried in the guts somewhere.

Works Cited

[1] Edmund Landau (1953) [first published 1909]. "$$\S$$56, Über Summen der Gestalt $$\sum _{p\le x}F(p,x)$$". Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, p. 211. Chelsea Publishing Company.

[2] J. M. De Koninck (1994). "On the Largest Prime Divisors of an Integer, Extreme Value Theory and Applications", pp. 447–462. Springer.

[3] Nathan McNew (2017). "The Most Frequent Values of the Largest Prime Divisor Function", pp. 210–224. Experimental Mathematics, Vol. 26, Issue 2.

[4] P. Erdős and L. Mirsky (1952). "The distribution of values of the divisor function $$d(n)$$". Proceedings of the London Mathematical Society, Volume s3-2, Issue 1, pp. 257–271.

• $M(x)$ will go to infinity as $x$ tends to infinity. Fixing $k \geq 1$, one can choose $x$ sufficiently large so that the number of numbers in $[1,x]$ with exactly $k$ prime factors is smaller than those with $k+1$ prime factors, by the result of Landau you mentioned. Indeed, Montgomery and Vaughan gives quite precise estimates and ranges of uniformity for the asymptotic formula. This shows that no fixed $k$ can be the mode for all $x$. Sep 25, 2019 at 22:21
• It's a much harder question to estimate $M(x)$ precisely (i.e., an asymptotic relation), since one would need to deal with cases where uniformity estimates are not available. Sep 25, 2019 at 22:21
• Rephrasing Stanley's comment in a streamlined form: the divergence of $M(x)$ follows from the two statements $d(n)\ge 2^{\omega(n)}$ (where $\omega(n)$ is the number of distinct prime factors of $n$) and "For any fixed $k$, the set of integers $n$ with $\omega(n)\le k$ has density $0$." Sep 25, 2019 at 22:25
• Thanks, I should've caught that! Sep 25, 2019 at 22:27
• A similar question was raised at m.se last year, math.stackexchange.com/questions/2774937/… Sep 25, 2019 at 23:06