  A Rb deficiency by a factor two with respect to the Sun has been found in M
dwarfs of solar metallicity. This deficiency is difficult to understand from
both the observational and nucleosynthesis point of views. To test the
reliability of this Rb deficiency, we study the Rb and Zr abundances in a
sample of KM-type giant stars in a similar metallicity range extracted from the
AMBRE Project. We derive Rb and Zr abundances in 54 giant stars with
metallicity close to solar by spectral synthesis in LTE and NLTE. The impact of
the Zeeman broadening in the RbI line is also studied. The LTE analysis results
in a Rb deficiency in giant stars smaller than that obtained in M dwarfs, but
the NLTE [Rb/Fe] ratios are very close to solar in the full metallicity range.
This contrasts with the figure found in M dwarfs. We investigate the effect of
gravitational settling and magnetic activity as possible causes of the Rb
deficiency found in M dwarfs. While, the former phenomenon has a negligible
impact on the surface Rb abundance, the existence of an average magnetic field
with intensity typical of that observed in M dwarfs may result in systematic Rb
abundance underestimations if the Zeeman broadening is not considered in the
spectral synthesis. The new [Rb,Zr/Fe] vs. [Fe/H] relationships can be
explained when the Rb production by rotating massive stars and
low-and-intermediate mass stars are considered, without the need of any
deviation from the standard s-process nucleosynthesis in AGB stars as
previously suggested.
