We are looking for real numbers $\alpha$ and $d$ for which there exist ``many'' real numbers $\tau$ such that the shifts of the Hurwitz-zeta function $\zeta(s+i\tau,\alpha)$ and $\zeta(s+id\tau,\alpha)$ are ``near'' each other.
"Self-approximation of Hurwitz Zeta-functions." Funct. Approx. Comment. Math. 51 (1) 181 - 188, September 2014. https://doi.org/10.7169/facm/2014.51.1.10