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

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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.

References

  1. P. Akhilesh, O. Ramaré, Explicit averages of non-negative multiplicative functions: going beyond the main term. Colloq. Math. 147, 275–313 (2017)

    Google Scholar 

  2. P. Akhilesh, O. Ramaré, Tail of a moebius sum with coprimality conditions. Integers 18, Paper No. A4, 6 (2018)

    Google Scholar 

  3. A. Axer, Beitrag zur Kenntnis der zahlentheoretischen Funktionen μ(n) und λ(n). Prace Matematyczno-Fizyczne 21(1), 65–95 (1910)

    Google Scholar 

  4. A. Axer, Über einige Grenzwertsätze. Wien. Ber. 120, 1253–1298 (1911) (German)

    Google Scholar 

  5. C. Axler, New bounds for the prime counting function. Integers 16 (2016), Paper No. A22, 15.

    Google Scholar 

  6. M. Balazard, Elementary remarks on Möbius’ function. Proc. Steklov Inst. Math. 276, 33–39 (2012)

    Google Scholar 

  7. H. Cohen, F. Dress, M. El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function. Funct. Approx. Comment. Math. 37(part 1), 51–63 (2007)

    Google Scholar 

  8. H. Davenport, On some infinite series involving arithmetical functions. Q. J. Math. Oxf. Ser. 8, 8–13 (1937)

    Google Scholar 

  9. H.G. Diamond, W.B. Zhang, A PNT equivalence for Beurling numbers. Funct. Approx. Comment. Math. 46(part 2), 225–234 (2012)

    Google Scholar 

  10. L.E. Dickson, Theory of Numbers (Chelsea Publishing Company, New York, 1971)

    Google Scholar 

  11. F. Dress, M. El Marraki, Fonction sommatoire de la fonction de Möbius 2. Majorations asymptotiques élémentaires. Exp. Math. 2(2), 89–98 (1993)

    Google Scholar 

  12. P. Dusart, Autour de la fonction qui compte le nombre de nombres premiers. Ph.D. thesis, Limoges, 1998, 173 pp., http://www.unilim.fr/laco/theses/1998/T1998_01.pdf

  13. P. Dusart, Estimates of some functions over primes without R. H. (2010), http://arxiv.org/abs/1002.0442

  14. M. El Marraki, Fonction sommatoire de la fonction μ de Möbius, majorations asymptotiques effectives fortes. J. Théor. Nombres Bordx. 7(2), 407–433 (1995)

    Google Scholar 

  15. I.C. Gohberg, M.G. Kreı̆n, Introduction to the Theory of Linear Nonselfadjoint Operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, vol. 18 (American Mathematical Society, Providence, 1969)

    Google Scholar 

  16. X. Gourdon, P. Demichel, The 1013 first zeros of the Riemann Zeta Function and zeros computations at very large height (2004), http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf

  17. J.P. Gram, Undersøgelser angaaende Maengden af Primtal under en given Graense. Résumé en français. Kjöbenhavn. Skrift. (6) II, 185–308 (1884)

    Google Scholar 

  18. A. Granville, O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 43(1), 73–107 (1996)

    Article  MathSciNet  Google Scholar 

  19. H. Halberstam, H.E. Richert, On a result of R. R. Hall. J. Number Theory 11, 76–89 (1979)

    Article  Google Scholar 

  20. H. Kadiri, A. Lumley, Short effective intervals containing primes. Integers 14 (2014), Paper No. A61, 18, arXiv:1407:7902

    Google Scholar 

  21. A. Kienast, Über die Äquivalenz zweier Ergebnisse der analytischen Zahlentheorie. Math. Ann. 95, 427–445 (1926), https://doi.org/10.1007/BF01206619

    Article  MathSciNet  Google Scholar 

  22. E. Landau, Neuer Beweis der Gleichung \(\sum _{k=1}^\infty \frac {\mu (k)}k=0\), Berlin. 16 S. gr. 8 (1899)

    Google Scholar 

  23. E. Landau, Über die Äquivalenz zweier Hauptsätze der analytischen Zahlentheorie. Wien. Ber. 120, 973–988 (1911) (German)

    Google Scholar 

  24. E. Landau, Über einige neuere Grenzwertsätze. Rendiconti del Circolo Matematico di Palermo (1884–1940) 34, 121–131 (1912), https://doi.org/10.1007/BF03015010

  25. B.V. Levin, A.S. Fainleib, Application of some integral equations to problems of number theory. Russ. Math. Surv. 22, 119–204 (1967)

    Article  Google Scholar 

  26. R.A. MacLeod, A curious identity for the Möbius function. Utilitas Math. 46, 91–95 (1994)

    MathSciNet  MATH  Google Scholar 

  27. E. Meissel, Observationes quaedam in theoria numerorum. J. Reine Angew. Math. 48, 301–316 (1854) (Latin)

    Google Scholar 

  28. L. Panaitopol, A formula for π(x) applied to a result of Koninck-Ivić. Nieuw Arch. Wiskd. (5) 1(1), 55–56 (2000)

    Google Scholar 

  29. J. Peetre, Outline of a scientific biography of Ernst Meissel (1826–1895). Hist. Math. 22(2), 154–178 (1995), https://doi.org/10.1006/hmat.1995.1015

    Article  MathSciNet  Google Scholar 

  30. D.J. Platt, Numerical computations concerning the GRH. Ph.D. thesis, 2013, http://arxiv.org/abs/1305.3087

  31. D.J. Platt, O. Ramaré, Explicit estimates: from Λ(n) in arithmetic progressions to Λ(n)∕n. Exp. Math. 26, 77–92 (2017)

    Google Scholar 

  32. O. Ramaré, Sur un théorème de Mertens. Manuscripta Math. 108, 483–494 (2002)

    Article  MathSciNet  Google Scholar 

  33. O. Ramaré, Arithmetical Aspects of the Large Sieve Inequality. Harish-Chandra Research Institute Lecture Notes, vol. 1 (Hindustan Book Agency, New Delhi, 2009). With the collaboration of D.S. Ramana

    Google Scholar 

  34. O. Ramaré, Explicit estimates for the summatory function of Λ(n)∕n from the one of Λ(n). Acta Arith. 159(2), 113–122 (2013)

    Article  MathSciNet  Google Scholar 

  35. O. Ramaré, From explicit estimates for the primes to explicit estimates for the Moebius function. Acta Arith. 157(4), 365–379 (2013)

    Article  MathSciNet  Google Scholar 

  36. O. Ramaré, Explicit estimates on the summatory functions of the Moebius function with coprimality restrictions. Acta Arith. 165(1), 1–10 (2014)

    Article  MathSciNet  Google Scholar 

  37. O. Ramaré, Explicit estimates on several summatory functions involving the Moebius function. Math. Comp. 84(293), 1359–1387 (2015)

    Article  MathSciNet  Google Scholar 

  38. O. Ramaré, Quantitative steps in Axer-Landau equivalence theorem (2017, submitted), 9pp.

    Google Scholar 

  39. J.B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)

    MathSciNet  MATH  Google Scholar 

  40. L. Schoenfeld, An improved estimate for the summatory function of the Möbius function. Acta Arith. 15, 223–233 (1969)

    Article  MathSciNet  Google Scholar 

  41. S. Selberg, Über die Summe \(\sum _{n\leqslant x}{\mu (n)\over nd(n)}\), in Tolfte Skandinaviska Matematikerkongressen, Lund (Lunds Universitets Matematiska Institution, Lund, 1954), pp. 264–272

    Google Scholar 

  42. D.W. Swann, Some new classes of kernels whose Fredholm determinants have order less than one. Trans. Am. Math. Soc. 160, 427–435 (1971)

    Article  MathSciNet  Google Scholar 

  43. T. Tao, A remark on partial sums involving the Möbius function. Bull. Aust. Math. Soc. 81(2), 343–349 (2010) (English)

    Article  MathSciNet  Google Scholar 

  44. The PARI Group, Bordeaux, PARI/GP, version 2.7.0 (2014), http://pari.math.u-bordeaux.fr/

  45. R. Vanlalngaia, Fonctions de Hardy des séries L et sommes de Mertens explicites. Ph.D. thesis, Mathématique, Lille (2015), http://math.univ-lille1.fr/~ramare/Epsilons/theseRamdinmawiaVanlalngaia.pdf

  46. W.B. Zhang, A generalization of Halász’s theorem to Beurling’s generalized integers and its application. Ill. J. Math. 31(4), 645–664 (1987)

    MATH  Google Scholar 

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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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