Non-real poles on the axis of absolute convergence of the zeta functions associated to Pascal's triangle modulo a prime

TL;DR

This paper extends previous work on meromorphic functions related to Pascal's triangle modulo a prime, demonstrating the existence of non-real poles on the axis of absolute convergence for a broader class of such functions.

Contribution

It generalizes Essouabri's results by considering alternative counting methods in Pascal's triangle modulo a prime, revealing new cases with non-real poles.

Findings

Existence of non-real poles on the axis of absolute convergence for extended classes of functions.

Generalization of previous results to different counting schemes in Pascal's triangle.

Connections to fractal geometry and meromorphic functions related to binomial coefficients.

Abstract

Picking binomial coefficients which cannot be divided by a given prime from Pascal's triangle, we find that they form a set with self-similarity. Essouabri studied on a class of meromorphic functions associated to the above set. These functions are related to fractal geometry and it is a problem whether such a function has a non-real pole on its axis of absolute convergence. Essouabri gave a proof of existence of such a non-real pole in the simplest case. The keys of his proof are Stein's and Wilson's estimates on how fast the points multiply in Pascal's triangle modulo a prime. This article will give an extension of Essouabri's result to some cases with certain ways to count the points in Pascal's triangle modulo a prime which are different from the traditional one.