A remark on joint approximation by one Dirichlet series

Abstract. We consider one absolutely convergent series ${\zeta }_{u_T}(s)$ connected to the Riemann zeta-function, and prove a theorem on joint approximation of analytic functions by shifts $({\zeta }_{u_T}(s+i{t}_{\tau }^{{\alpha }_1}),\dots ,{\zeta }_{u_T}(s+i{t}_{\tau }^{{\alpha }_r}))$ as $T\to \mathrm{\infty }$, where ${\alpha }_1,\dots ,{\alpha }_r$ are fixed different positive numbers, $t_{\tau }$ is the Gram function, and $u_T\to \mathrm{\infty }$ and $u_T\ll T^2$ as $T\to \mathrm{\infty }$.