The impact of relativistic effects on the 3D Quasar-Lyman-$\alpha$
  cross-correlation
  We study the impact of relativistic effects in the 3-dimensional
cross-correlation between Lyman-$\alpha$ forest and quasars. Apart from the
relativistic effects, which are dominated by the Doppler contribution, several
systematic effects are also included in our analysis (intervening metals,
unidentified high column density systems, transverse proximity effect and
effect of the UV fluctuations). We compute the signal-to-noise ratio for the
Baryonic Oscillation Spectroscopic Survey (BOSS), the extended Baryonic
Oscillation Spectroscopic Survey (eBOSS) and the Dark Energy Spectroscopic
Instrument (DESI) surveys, showing that DESI will be able to detect the Doppler
contribution in a Large Scale Structure (LSS) survey for the first time, with a
S/N $>7$ for $r_{\rm min} > 10$ Mpc$/h$, where r$_{\rm min}$ denotes the
minimum comoving separation between sources. We demonstrate that several
physical parameters, introduced to provide a full modelling of the
cross-correlation function, are affected by the Doppler contribution. By using
a Fisher matrix approach, we establish that if the Doppler contribution is
neglected in the data analysis, the derived parameters will be shifted by a
non-negligible amount for the upcoming surveys.
