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.
If only by the fact that they define the same function!
- 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.
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.
The Beurling integers are the multiplicative semi-group built on a family of “primes” to be chosen real numbers from (1, ∞).
- 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.
I formulated a more precise conjecture, say Conjecture B, in [35].
- 7.
As already stated, we show that by classically repeated integrations by parts.
- 8.
F. Daval was at the time a PhD student of mine.
- 9.
As a matter of fact, the mentioned lemma is slightly different, but a corrigendum is on its way.
- 10.
This one is Atle Selberg!
- 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
P. Akhilesh, O. Ramaré, Explicit averages of non-negative multiplicative functions: going beyond the main term. Colloq. Math. 147, 275–313 (2017)
P. Akhilesh, O. Ramaré, Tail of a moebius sum with coprimality conditions. Integers 18, Paper No. A4, 6 (2018)
A. Axer, Beitrag zur Kenntnis der zahlentheoretischen Funktionen μ(n) und λ(n). Prace Matematyczno-Fizyczne 21(1), 65–95 (1910)
A. Axer, Über einige Grenzwertsätze. Wien. Ber. 120, 1253–1298 (1911) (German)
C. Axler, New bounds for the prime counting function. Integers 16 (2016), Paper No. A22, 15.
M. Balazard, Elementary remarks on Möbius’ function. Proc. Steklov Inst. Math. 276, 33–39 (2012)
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)
H. Davenport, On some infinite series involving arithmetical functions. Q. J. Math. Oxf. Ser. 8, 8–13 (1937)
H.G. Diamond, W.B. Zhang, A PNT equivalence for Beurling numbers. Funct. Approx. Comment. Math. 46(part 2), 225–234 (2012)
L.E. Dickson, Theory of Numbers (Chelsea Publishing Company, New York, 1971)
F. Dress, M. El Marraki, Fonction sommatoire de la fonction de Möbius 2. Majorations asymptotiques élémentaires. Exp. Math. 2(2), 89–98 (1993)
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
P. Dusart, Estimates of some functions over primes without R. H. (2010), http://arxiv.org/abs/1002.0442
M. El Marraki, Fonction sommatoire de la fonction μ de Möbius, majorations asymptotiques effectives fortes. J. Théor. Nombres Bordx. 7(2), 407–433 (1995)
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)
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
J.P. Gram, Undersøgelser angaaende Maengden af Primtal under en given Graense. Résumé en français. Kjöbenhavn. Skrift. (6) II, 185–308 (1884)
A. Granville, O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 43(1), 73–107 (1996)
H. Halberstam, H.E. Richert, On a result of R. R. Hall. J. Number Theory 11, 76–89 (1979)
H. Kadiri, A. Lumley, Short effective intervals containing primes. Integers 14 (2014), Paper No. A61, 18, arXiv:1407:7902
A. Kienast, Über die Äquivalenz zweier Ergebnisse der analytischen Zahlentheorie. Math. Ann. 95, 427–445 (1926), https://doi.org/10.1007/BF01206619
E. Landau, Neuer Beweis der Gleichung \(\sum _{k=1}^\infty \frac {\mu (k)}k=0\), Berlin. 16 S. gr. 8∘ (1899)
E. Landau, Über die Äquivalenz zweier Hauptsätze der analytischen Zahlentheorie. Wien. Ber. 120, 973–988 (1911) (German)
E. Landau, Über einige neuere Grenzwertsätze. Rendiconti del Circolo Matematico di Palermo (1884–1940) 34, 121–131 (1912), https://doi.org/10.1007/BF03015010
B.V. Levin, A.S. Fainleib, Application of some integral equations to problems of number theory. Russ. Math. Surv. 22, 119–204 (1967)
R.A. MacLeod, A curious identity for the Möbius function. Utilitas Math. 46, 91–95 (1994)
E. Meissel, Observationes quaedam in theoria numerorum. J. Reine Angew. Math. 48, 301–316 (1854) (Latin)
L. Panaitopol, A formula for π(x) applied to a result of Koninck-Ivić. Nieuw Arch. Wiskd. (5) 1(1), 55–56 (2000)
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
D.J. Platt, Numerical computations concerning the GRH. Ph.D. thesis, 2013, http://arxiv.org/abs/1305.3087
D.J. Platt, O. Ramaré, Explicit estimates: from Λ(n) in arithmetic progressions to Λ(n)∕n. Exp. Math. 26, 77–92 (2017)
O. Ramaré, Sur un théorème de Mertens. Manuscripta Math. 108, 483–494 (2002)
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
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)
O. Ramaré, From explicit estimates for the primes to explicit estimates for the Moebius function. Acta Arith. 157(4), 365–379 (2013)
O. Ramaré, Explicit estimates on the summatory functions of the Moebius function with coprimality restrictions. Acta Arith. 165(1), 1–10 (2014)
O. Ramaré, Explicit estimates on several summatory functions involving the Moebius function. Math. Comp. 84(293), 1359–1387 (2015)
O. Ramaré, Quantitative steps in Axer-Landau equivalence theorem (2017, submitted), 9pp.
J.B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)
L. Schoenfeld, An improved estimate for the summatory function of the Möbius function. Acta Arith. 15, 223–233 (1969)
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
D.W. Swann, Some new classes of kernels whose Fredholm determinants have order less than one. Trans. Am. Math. Soc. 160, 427–435 (1971)
T. Tao, A remark on partial sums involving the Möbius function. Bull. Aust. Math. Soc. 81(2), 343–349 (2010) (English)
The PARI Group, Bordeaux, PARI/GP, version 2.7.0 (2014), http://pari.math.u-bordeaux.fr/
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
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)
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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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