"[Consider] the function zeta(z) = Trace((I-zP)^(-1)) where P is a stochastic matrix (i.e., a matrix of non-negative reals with the entries in each row summing to 1). Expanding zeta(z) as Trace(I) + z Trace(P) + z^2 Trace(P^2) + ..., one can see that this is a sum of closed orbits in the state-space of the Markov chain associated with the matrix P, where the weight of a closed orbit of length n is z^n times the product of the probabilities going around the orbit. So it's a sort of dynamical zeta-function. If one regularizes the trace at z=1 by subtracting off 1/(1-z), one obtains a spectral invariant of P called Kemeny's constant, which measures how "big" the Markov chain is. It has a nice probabilistic interpretation; see math.dartmouth.edu/~doyle/docs/kc/kc.pdf. So I was wondering if other aspects of zeta(z) have been studied (e.g., if we differentiate it, does the regularized value at z=1 have a probabilistic interpretation? do the zeroes of the zeta(z) obey any patterns?)" email: prop 'at' jamespropp 'dot' org