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Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play

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Irregularities in the Distribution of Prime Numbers

Abstract

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field \(\mathbb Q(\zeta _q)\) with q a prime. Kummer’s conjecture concerns the asymptotic behaviour of the first factor of the class number of \(\mathbb Q(\zeta _q)\) and Ihara’s the positivity of the Euler-Kronecker constant of \(\mathbb Q(\zeta _q)\) (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function \(\zeta _{\mathbb Q(\zeta _q)}(s)\) at s = 1). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes.

With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.

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Notes

  1. 1.

    In fact, the title of this paper ends with a question mark. Since it is considered very bad style to have it in the title of a paper, this footnote might be a better place. Not putting the question mark would go against the moral of this paper.

  2. 2.

    The similarity was first noted by Andrew Granville, see acknowledgment.

  3. 3.

    I have not come across this formula in the literature.

  4. 4.

    The authors of [7], unaware of Granville’s work and the fact that they were dealing with a(n) (ex-)conjecture of Erdős, gave a short different proof using a 1961 paper of…Erdős [6] himself. (The title of [6] has “G. Golomb” instead of the correct “S. Golomb”.)

  5. 5.

    He assumes GRH. The reproof given in [7, p. 1470] shows that ERH is sufficient.

  6. 6.

    Having the larger error term o(L 3) would also suffice for our purposes.

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Acknowledgements

I would like to thank James Maynard for pointing out that one can take C = 1∕246 in Challenge 1. Furthermore, I am grateful to Alexandru Ciolan, Sumaia Saad Eddin and Alisa Sedunova for proofreading and help with editing an earlier version. Ignazio Longhi and the referee kindly pointed out some disturbing typos.

The similarity between Kummer’s and Ihara’s conjectures was pointed out by Andrew Granville after a talk given by Kevin Ford on [7]. At that point the authors of [7] had independently obtained Theorem 1, but not Granville’s Proposition 1, the latter being precisely the result used by Granville to unleash the \(\log \log \log \) devil. Once at the loose, it created havoc also among the Euler-Kronecker constants.

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  1. Max Planck Institute for Mathematics, Bonn, Germany

    Pieter Moree

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  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    János Pintz

  2. Institute of Mathematics, University of Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Moree, P. (2018). Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play. In: Pintz, J., Rassias, M. (eds) Irregularities in the Distribution of Prime Numbers. Springer, Cham. https://doi.org/10.1007/978-3-319-92777-0_8

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