Skip to main content
Log in

Asymptotic formula of the number of Newton polygons

Mathematische Zeitschrift Aims and scope Submit manuscript

Cite this article


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2


  1. Báez-Duarte, L.: Hardy-Ramanujan’s asymptotic formula for partitions and the central limit theorem. Adv. Math. 125, 114–120 (1997)

    Article  MathSciNet  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  Google Scholar 

  3. Cannon, J., et al.: Magma a Computer Algebra System. School of Mathematics and Statistics, University of Sydney (2017).

  4. Cramér, H.: Random variables and probability distributions. In: Cambridge Tracts in Mathematics and Mathematical Physics, vol. 36, 3rd edn. Cambridge University Press, London-New York (1970)

  5. Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann. 212(3), 215–248 (1975)

    Article  MathSciNet  Google Scholar 

  6. Hardy, G.H., Ramanujan, S.: Asymptotic formulaæ in combinatory analysis. Proc. Lond. Math. Soc. S2–17(1), 75–115 (1918)

    Article  Google Scholar 

  7. Manin, Yu. I.: The theory of commutative formal groups over fields of finite characteristic. Russ. Math. Surv. 18, 1–80 (1963)

  8. Meinardus, G.: Asymptotische Aussagen über Partitionen. Math. Z. 59, 388–398 (1954)

    Article  MathSciNet  Google Scholar 

  9. Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie(1859)

  10. Titchmarsh, E.C.: The theory of the Riemann zeta-function, 2nd edn. The Clarendon Press, Oxford University Press, New York (1986). Edited and with a preface by D. R. Heath-Brown

    MATH  Google Scholar 

  11. Maple User Manual: Toronto: Maplesoft, a division of Waterloo Maple Inc. (2016).

  12. Computation programs and log files for the paper: “Asymptotic formula of the number of Newton polygons”.

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Shushi Harashita.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harashita, S. Asymptotic formula of the number of Newton polygons. Math. Z. 297, 113–132 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification