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M-dwarf stars are promising targets for identifying and characterizingpotentially habitable planets. K2-3 is a nearby (45 pc), early-type M dwarfhosting three small transiting planets, the outermost of which orbits close tothe inner edge of the stellar (optimistic) habitable zone. The K2-3 system iswell suited for follow-up characterization studies aimed at determiningaccurate masses and bulk densities of the three planets. Using a total of 329radial velocity measurements collected over 2.5 years with the HARPS-N andHARPS spectrographs and a proper treatment of the stellar activity signal, weaim to improve measurements of the masses and bulk densities of the K2-3planets. We use our results to investigate the physical structure of theplanets. We analyse radial velocity time series extracted with two independentpipelines by using Gaussian process regression. We adopt a quasi-periodickernel to model the stellar magnetic activity jointly with the planetarysignals. We use Monte Carlo simulations to investigate the robustness of ourmass measurements of K2-3\,c and K2-3\,d, and to explore how additionalhigh-cadence radial velocity observations might improve them. Despite thestellar activity component being the strongest signal present in the radialvelocity time series, we are able to derive masses for both planet b ($M_{\rmb}=6.6\pm1.1$ $M_{\rm \oplus}$) and planet c ($M_{\rm c}=3.1^{+1.3}_{-1.2}$$M_{\rm \oplus}$). The Doppler signal due to K2-3\,d remains undetected, likelybecause of its low amplitude compared to the radial velocity signal induced bythe stellar activity. The closeness of the orbital period of K2-3\,d to thestellar rotation period could also make the detection of the planetary signalcomplicated. [...]

Eyes on K2-3: A system of three likely sub-Neptunes characterized with HARPS-N and HARPS