  To obtain the Dirichlet series for complex powers of the Riemann zeta
function, we define and study the basic properties of a sequence of polynomials
that, used as coefficients of the respective terms of the Dirichlet series of
the Riemann zeta function in the half plane $x > 1$, produces the required
exponential function. Unlike the method described in ([4], p.~278), which
requires more advanced knowledge of the relationships between Dirichlet series
and multiplicative arithmetic functions, our approach only needs mathematical
induction on the total number of prime divisors of $n$, the Dirichlet product
and the use of an analytic property characteristic of the exponential function
in the complex plane.
