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Abstract: A graph theoretical analog of Brauer–Siegel theory for zeta functions of number fields is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph. [Copyright &y& Elsevier]

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x2+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for t=[√ of m] and consecutive integers x=x0, x0+1,…,x0+t−1, | f(x)| is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials fm(x) such that 1+4m is square free. In this note, we shall remove the condition that 1+4m is square free. [Copyright &y& Elsevier]