In this paper, we enumerate Newton polygons asymptotically. The number of Newton polygons is computable by a simple recurrence equation, but unexpectedly the asymptotic formula of its logarithm contains growing oscillatory terms. As the terms come from non-trivial zeros of the Riemann zeta function, an estimation of the amplitude of the oscillating part is equivalent to the Riemann hypothesis.
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This work started with the graduation thesis (February in 2017) by Takuya Tani (supervised by the author), where a variant of the generating function (2) was found. I would like to thank Professor Norio Konno for helpful suggestions and supports continuing after Tani’s presentation of his graduation thesis, which led me to the probabilistic approach. The author also thank the anonymous referee for his/her careful reading and helpful comments. This work was supported by JSPS Grant-in-Aid for Scientific Research (C) 17K05196.
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Harashita, S. Asymptotic formula of the number of Newton polygons. Math. Z. 297, 113–132 (2021). https://doi.org/10.1007/s00209-020-02504-w