On the RR Lyrae stars in globulars: V. the complete Near-Infrared (JHKs) census of omega Centauri RR Lyrae variables.

We present a new complete Near-Infrared (NIR, $JHK_s$) census of RR Lyrae stars (RRLs) in the globular $\omega$ Cen (NGC 5139). We collected 15,472 $JHK_s$ images with 4-8m class telescopes over 15 years (2000-2015) covering a sky area around the cluster center of 60x34 arcmin$^2$. These images provided calibrated photometry for 182 out of the 198 cluster RRL candidates with ten to sixty measurements per band. We also provide new homogeneous estimates of the photometric amplitude for 180 ($J$), 176 ($H$) and 174 ($K_s$) RRLs. These data were supplemented with single-epoch $JK_s$ magnitudes from VHS and with single-epoch $H$ magnitudes from 2MASS. Using proprietary optical and NIR data together with new optical light curves (ASAS-SN) we also updated pulsation periods for 59 candidate RRLs. As a whole, we provide $JHK_s$ magnitudes for 90 RRab (fundamentals), 103 RRc (first overtones) and one RRd (mixed--mode pulsator). We found that NIR/optical photometric amplitude ratios increase when moving from first overtone to fundamental and to long-period (P>0.7 days) fundamental RRLs. Using predicted Period-Luminosity-Metallicity relations, we derive a true distance modulus of 13.674$\pm$0.008$\pm$0.038 mag (statistical error and standard deviation of the median)---based on spectroscopic iron abundances---and of 13.698$\pm$0.004$\pm$0.048 mag---based on photometric iron abundances. We also found evidence of possible systematics at the 5-10% level in the zero-point of the PLs based on the five calibrating RRLs whose parallaxes had been determined with HST