An analog of the prime number theorem for finite fields via truncated polylogarithm expansions

An exponentially accurate asymptotic expansion of the truncated polylogarithm function is derived that leads to an asymptotic formula for enumerating monic irreducible polynomials over finite fields. This formula is analogous to the asymptotic expansion formula of the classical prime counting function. Results are presented which show that it is more accurate than previous results in the literature while requiring very little computational effort. Asymptotic expansions of the Lerch transcendent, Eulerian polynomials, and polylogarithms of negative integer order are also given. The accuracy of the proposed approach is verified via numerical results.