Signaling to Relativistic Observers: An Einstein-Shannon-Riemann Encounter

A communication scenario is described involving a series of events triggered by a transmitter and observed by a receiver experiencing relativistic time dilation. The message selected by the transmitter is assumed to be encoded in the events' timings and is required to be perfectly recovered by the receiver, regardless of the difference in clock rates in the two frames of reference. It is shown that the largest proportion of the space of all $k$-event signals that can be selected as a code ensuring error-free information transfer in this setting equals $\zeta(k)^{-1}$, where $\zeta$ is the Riemann zeta function.