Meromorphic continuation of multivariable Euler products
Gautami Bhowmik, Driss Essouabri, Ben Lichtin · Forum Mathematicum · 2007
This article extends classical one variable results about Euler products, defined by integral valued polynomial or analytic functions, to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and give a criterion, à la Estermann-Dahlquist, for the existence of a meromorphic extension to . In addition, we precisely describe the boundaries of analyticity and meromorphy for a multivariable Euler product determined by any toric variety (split over ). Using our method, we are also able to calculate a precise asymptotic for the number of n -fold products of integers that equal the n th power of an integer, for any n ≥ 3.