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Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics pp 293–315Cite as

On the Missing Log Factor

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2213))

Abstract

This is a written account of a talk with the same title given during the main Chaire Morlet conference. Its guideline is the elementarily proven bound \(|\sum _{n\leqslant x}\mu (n)/n|\leqslant 1\). The trivial bound for the implied summation is \(\log x+ \operatorname {\mathrm {\mathcal {O}}}(1)\), while the Prime Number Theorem tells us that it is o(1). Our starting estimate thus lies in-between, a fact that we explore under different lights.

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Notes

  1. 1.

    If only by the fact that they define the same function!

  2. 2.

    His full name is Daniel Friedrich Ernst Meissel. This student of Carl Gustav Jacob Jacobi and Johann Peter Gustav Lejeune Dirichlet was born in 1826 and passed away in 1895. His full biography can be found in [29].

  3. 3.

    This reference has been kindly provided to us by M. Balazard. The reader is referred to the MacTutor archive maintained by the University of Saint Andrew, in Scotland for the biography of J.-P. Gram. We just mention here that Meissel travelled to Denmark in 1885 to meet the 23 years old Gram who had just won the Gold Medal of the Royal Danish Academy of Sciences for the memoir we refer to. The inequality we extract from this memoir is not its main matter but rather a pleasant side dish.

  4. 4.

    The Beurling integers are the multiplicative semi-group built on a family of “primes” to be chosen real numbers from (1, ∞).

  5. 5.

    This computation has not been the subject of any published paper. D. Platt in [30] has checked this hypothesis up to height 109 by with a very precise program using interval arithmetic.

  6. 6.

    I formulated a more precise conjecture, say Conjecture B, in [35].

  7. 7.

    As already stated, we show that by classically repeated integrations by parts.

  8. 8.

    F. Daval was at the time a PhD student of mine.

  9. 9.

    As a matter of fact, the mentioned lemma is slightly different, but a corrigendum is on its way.

  10. 10.

    This one is Atle Selberg!

  11. 11.

    The Liouville function is the completely multiplicative function defined by λ(n) = (−1)Ω(n), where Ω(n) is the number of prime factors of n, counted with multiplicity, so that Ω(12) = 3.

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Authors and Affiliations

  1. Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, I2M – UMR 7373, Marseille, France

    Olivier Ramaré

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  1. Olivier Ramaré
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Editors and Affiliations

  1. CNRS UMR 7373, Institut de Mathématiques de Marseille, Marseille, France

    Sébastien Ferenczi

  2. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Joanna Kułaga-Przymus

  3. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Mariusz Lemańczyk

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Ramaré, O. (2018). On the Missing Log Factor. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_15

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