The HARPS search for southern extra-solar planets XXXI. The M-dwarf sample

Context. Searching for planets around stars with different masses helps us to assess the outcome of planetary formation for different initial conditions. The low-mass M dwarfs are also the most frequent stars in our Galaxy and potentially therefore, the most frequent planet hosts. Aims We present observations of 102 southern nearby M dwarfs, using a fraction of our guaranteed time on the ESO/HARPS spectrograph. We observed for 460 h and gathered 1965 precise (similar to 1-3 m/s) radial velocities (RVs), spanning the period from Feb. 11, 2003 to Apr. 1, 2009. Methods. For each star observed, we derive a time series and its precision as well as its variability. We apply systematic searches for long-term trends, periodic signals, and Keplerian orbits (from one to four planets). We analyze the subset of stars with detected signals and apply several diagnostics to discriminate whether the observed Doppler shifts are caused by either stellar surface inhomogeneities or the radial pull of orbiting planets. To prepare for the statistical view of our survey, we also compute the limits on possible unseen signals, and derive a first estimate of the frequency of planets orbiting M dwarfs. Results. We recover the planetary signals of 9 planets announced by our group (Gl 176 b, Gl 581 b, c, d & e, Gl 674 b, Gl 433 b, Gl 667C b, and Gl 667C c). We present radial velocities confirming that GJ 849 hosts a Jupiter-mass planet, plus a long-term radial-velocity variation. We also present RVs that precise the planetary mass and period of Gl 832b. We detect long-term RV changes for Gl 367, Gl 680, and Gl 880, which are indicative of yet unknown long-period companions. We identify candidate signals in the radial-velocity time series of 11 other M dwarfs. Spectral diagnostics and/or photometric observations demonstrate however that these signals are most probably caused by stellar surface inhomogeneities. Finally, we find that our survey is sensitive to a few Earth-mass planets for periods up to several hundred days. We derive a first estimate of the occurrence of M-dwarf planets as a function of their minimum mass and orbital period. In particular, we find that giant planets (m sin i = 100-1000 M-circle plus) have a low frequency (e.g. integral less than or similar to 1 % for P = 1-10 d and integral = 0.02(-0.01)(+0.03) for P = 10-100 d), whereas super-Earths (m sin i = 1-10 M-circle plus) are likely very abundant (integral = 0.36(-0.10)(+0.25) for P = 1-10 d and integral = 0.52(-0.16)(+0.50) for P = 10-100 d). We also obtained eta(circle plus) = 0.41-0.13,+0.54 which is the frequency of habitable planets orbiting M dwarfs (1 <= m sin i <= 10 M-circle plus). For the first time. eta(circle plus) is a direct measure and not a number extrapolated from the statistics of more massive and/or shorter-period planets.