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On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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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 lea …