Liang Zhang

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

Assume that the Riemann zeta function has no zeros in $\Re(s)>1-\delta$, $s\neq1$, for some fixed $0<\delta\leq1/2$. We prove that, for every sufficiently large integer $k$, a positive multiple $m_k k!$ has arithmetic complexity $\tau(m_k k!)\leq\exp(C\sqrt{\log k\log\log k})$, where $C$ depends only on $\delta$. We fi...