  (Abridged) We carried out a period analysis on ~3700 RR Lyrae stars spread
over ~50 square degrees towards the Sagittarius dwarf galaxy (Sgr). The RR
Lyraes are distributed almost evenly between Sgr and the Milky Way. For Sgr
members, the average periods are 0.574 day and 0.322 day for RRab and RRc stars
respectively. This places Sgr in the long-period tail of the Oosterhoff I
group. We report the detection of 53 double-mode RR Lyrae stars (RRd) within
our sample. The magnitude of 40 of these stars is consistent with membership in
Sgr whereas 13 RRds are located within our Galaxy. We also found 13 RR Lyraes
(5 in Sgr and 8 in the Galaxy) exhibiting two closely spaced frequencies, most
probably related to non-radial pulsations. The period distribution of the RR
Lyrae variables in Sgr is compared to those of other Milky Way satellites. We
find a remarkable similarity between the RR Lyrae populations in Sgr and the
Large Magellanic Cloud (LMC), suggesting that these galaxies have similar
horizontal branch morphologies. This may indicate that Sgr and the LMC started
their formation under similar conditions. Using various photometric indicators,
we estimate the metallicity of the RR Lyrae stars in Sgr and find <[Fe/H]>~-1.6
dex with a dispersion of ~ +/-0.5 dex around this value and a minor but
significant population at <-2.0 dex. We do not find evidence for a spatial
metallicity gradient in the RR Lyrae population of Sgr. From the spatial
distribution of RR Lyraes, we estimate the RRab content and total luminosity of
this galaxy and find MV(Sgr)~-14.7 mag, a value that would be consistent with
the empirical metallicity/luminosity relation for Dsph galaxies.
