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

Volume 178 / 2017

Tomohiro Ikkai Acta Arithmetica 178 (2017), 43-55 MSC: Primary 11M41; Secondary 11B65. DOI: 10.4064/aa8359-9-2016 Published online: 6 March 2017


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.


  • Tomohiro IkkaiGraduate School of Mathematics Nagoya University 464-8602 Nagoya, Japan

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