  We formulate a parametrized uniformly absolutely globally convergent series
of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by
Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the
polylogarithm function. As an immediate first application of the new
parametrized series, a new expression of $\zeta$(s) follows: (s -- 1)$\zeta$(s)
= -- 1 0 Li s z z -- 1 dz. As a second important application, using the
functional equation and exploiting uniform convergence of the series defining
Z(s, x), we have for any non-trivial zero s

Number Theory