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?

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.