Square-root logarithmic bounds for the arithmetic complexity of factorial multiples

Assume that the Riemann zeta function has no zeros in $\Re(s)>1-δ$, $s\neq1$, for some fixed $0<δ\leq1/2$. We prove that, for every sufficiently large integer $k$, a positive multiple $m_k k!$ has arithmetic complexity $τ(m_k k!)\leq\exp(C\sqrt{\log k\log\log k})$, where $C$ depends only on $δ$. We first establish a smooth-number asymptotic at every sufficiently large center $x$ in intervals of length at least $x^{3/4}$. Combining this estimate with an established genus-two construction yields the arithmetic bound without Cheng's conjecture on smooth numbers in elliptic Hasse intervals. The bound includes the cost of constructing all integer parameters. This work was carried out with assistance from the UniMath model, developed by UniDT (Shanghai) Co., Ltd., and OpenAI's GPT-6.1 Sol and GPT-6 Astra.