  This paper investigates a new family of special functions referred to as
hypergeometric zeta functions. Derived from the integral representation of the
classical Riemann zeta function, hypergeometric zeta functions exhibit many
properties analogous to their classical counterpart, including the intimate
connection to Bernoulli numbers. These new properties are treated in detail and
are used to demonstrate a functional inequality satisfied by second-order
hypergeometric zeta functions.
