Non-real poles on the axis of absolute convergence of the zeta functions associated to Pascal’s triangle modulo a prime
In Pascal’s triangle, the binomial coefficients not divisible by a given prime form a set with self-similarity. Essouabri studied a class of meromorphic functions associated to that set. These functions are related to fractal geometry and it is of interest whether such a function has a non-real pole on its axis of absolute convergence.
Essouabri proved the existence of such a pole in the simplest case. The keys to his proof are Stein’s and Wilson’s estimates on how fast the points multiply in the above self-similar set. This article gives an extension of Essouabri’s result.