  The three-year WMAP(WMAP3), combined with other cosmological observations
from galaxy clustering and Type Ia Supernova (SNIa), prefers a non-vanishing
running of the primordial spectral index independent of the low CMB multipoles.
Motivated by this feature we study cosmological constraint on the neutrino
mass, which severely depends on what prior we adopt for the spectral shape of
primordial fluctuations, taking possible running into account. As a result we
find a more stringent constraint on the sum of the three neutrino masses, m_\nu
< 0.76eV (2 \sigma), compared with m_\nu < 0.90eV (2 \sigma) for the case where
power-law prior is adopted to the primordial spectral shape.
