  We consider analytic functions of the Riemann zeta type, for which, if $s$ is
a zero, so is $1-s$. We use infinite product representations of these
functions, assuming their zeros to be of first order. We use exponential
factors to accelerate convergence, and by comparing the exponential factors
with the Taylor series coefficients about $s=0$ of the original function,
connect these coefficients with sums of powers of reciprocals of the zeros, in
the form of sum rules. Such sum rules have been previously considered by Lehmer
and Keiper, but the approach and applications taken here are more general. In
related work, a new sufficient condition is found for the Riemann hypothesis,
and the basis for this condition is discussed.
