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arXiv · 1509.04408

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

Tomohiro Ikkai
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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.

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Tomohiro Ikkai. 2015-09-15. Non-real poles on the axis of absolute convergence of the zeta functions associated to Pascal's triangle modulo a prime. https://arxiv.org/abs/1509.04408

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