Extend the uniform smooth multiplication-table asymptotic

Determine how large the smoothness parameter y=y(N) can be while the asymptotic relation M_k(N,y)\sim\Psi(N^k,y) continues to hold uniformly for every dimension k\geq 1, where M_k(N,y) counts products of k integers at most N whose prime factors are all at most y, and \Psi(N^k,y) counts y-smooth integers up to N^k.

Background

The paper defines M_k(N,y) as the number of distinct products x_1\cdots x_k with each x_i\leq N and P{+}(x_i)\leq y. It proves that when y=o(\log N), this quantity satisfies M_k(N,y)\sim\Psi(Nk,y) uniformly for all k\geq 1.

The authors then ask for the maximal growth range of y beyond y=o(\log N) in which the same uniform asymptotic remains valid. This is a concrete extension problem concerning the interaction between smooth-number counts and products formed from individually bounded smooth factors.

References

Theorem \ref{smooth_thm} naturally raises the question of how far the range of the smoothness parameter can be extended.

How large can $y=y(N)$ be while

M_k(N,y)\sim \Psi(Nk,y)

continues to hold uniformly for $k\geq 1$?

— The multiplication table problem in large dimensions  (2608.16163 - Sabuncu et al., 17 Aug 2026) in Section 1, subsection “A smooth variant,” immediately after Theorem 1 (Theorem \ref{smooth_thm}); displayed Problem environment