  Observing a multiple version of the divisor function we introduce a new zeta
function which we call a multiple finite Riemann zeta function. We utilize some
$q$-series identity for proving the zeta function has an Euler product and
then, describe the location of zeros. We study further multi-variable and
multi-parameter versions of the multiple finite Riemann zeta functions and
their infinite counterparts in connection with symmetric polynomials and some
arithmetic quantities called powerful numbers.
