# 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 , 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  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  to $$O(1)$$.)

Moreover, there is a paper by Erdős and Mirsky  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

 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.

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

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

 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