Abstract
This chapter is combinatorial and rather rewarding: we are going to present three procedures that yield interesting formulas in a rather magical way.
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Notes
- 1.
Thanks to the DigiZeitschriften project hosted by the university of Göttingen, we can have an access to this text online, though some knowledge of Latin is required. The classical reference book [4] on history of numbers of L.E. Dickson may serve as a first guide, and for instance, the paper [9] is mentioned in Chap. XIX of this series of three books.
- 2.
Gram won the Gold Medal of the Royal Danish Academy of Sciences in 1884 for the paper he published that contains inter alia this inequality, see [5, p 196-197].
References
R.P. Brent and J. van de Lune. “A note on Pólya’s observation concerning Liouville’s function”. In: Leven met getallen : liber amicorum ter gelegenheid van de pensionering van Herman te Riele. Ed. by J.A.J. van Vonderen (Coby). CWI (cit. on p. 58).
F. Daval. “Identités intégrales et estimations explicites associées pour les fonctions sommatoires liées á la fonction de Möbius et autres fonctions arithmétiques”. PhD thesis. Université de Lille, Mathematics, 2019 (cit. on pp. 60, 62).
H. Davenport. “On some infinite series involving arithmetical functions”. In: Quart. J. Math., Oxf. Ser. 8 (1937), pp. 8–13 (cit. on p. 59).
L.E. Dickson. Theory of numbers. Chelsea Publishing Company, 1971 (cit. on p. 59).
J.P. Gram. Undersøgelser angaaende Maengden af Primtal under en given Graense. Résumé en français. Danish. Kjöbenhavn. Skrift. (6) II. 185-308 (1884). 1884 (cit. on p. 59).
A.J. Granville and O. Ramaré. “Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients”. In: Mathematika 43.1 (1996), pp. 73–107 (cit. on p. 59).
K. Knopp. “Über Lambertsche Reihen.” In: J. Reine Angew. Math. 142 (1913), pp. 283–315 (cit. on p. 58).
J.H. Lambert. Anlage zur Architectonik oder Theorie des Einfachen und Ersten in des philosophischen und mathematischen Erkenntnis. 2 vol. Riga, 1771 (cit. on p. 58).
D.F.E. Meissel. “Observationes quaedam in theoria numerorum”. Latin. In: J. Reine Angew. Math. 48 (1854), pp. 301–316. https://doi.org/10.1515/crll.1854.48.301 (cit. on p. 59).
Ch.H. Müntz. “Beziehungen der Riemannschen \(\zeta \)-Funktion zu willkürlichen reellen Funktionen”. German. In: Mat. Tidsskr. B 1922 (1922), pp. 39–47 (cit. on p. 61).
G. Pólya and G. Szegő. Problems and theorems in analysis. II. Classics in Mathematics. Theory of functions, zeros, polynomials, determinants, number theory, geometry, Translated from the German by C. E. Billigheimer, Reprint of the 1976 English translation. Springer-Verlag, Berlin, 1998, pp. xii+392. https://doi.org/10.1007/978-3-642-61905-2_7 (cit. on p. 58).
O. Ramaré “Explicit estimates on several summatory functions involving the Moebius function”. In: Math. Comp. 84.293 (2015), pp. 1359–1387 (cit. on pp. 62, 63).
O. Ramaré. “Explicit average orders: news and problems”. In: Number theory week 2017. Vol. 118. Banach Center Publ. Polish Acad. Sci. Inst. Math., Warsaw, 2019, pp. 153–176 (cit. on pp. 61, 63).
G.-C. Rota. “On the foundations of combinatorial theory. I. Theory of Möbius functions”. In: Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). https://doi.org/10.1007/BF00531932 (cit. on p. 57).
T. Tao. “A remark on partial sums involving the Möbius function”. English. In: Bull. Aust. Math. Soc. 81.2 (2010), pp. 343–349. https://doi.org/10.1017/S0004972709000884 (cit. on p. 59).
E.C. Titchmarsh. The theory of the Riemann zeta-function. Second. Edited and with a preface by D.R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, 1986, pp. x+412 (cit. on p. 61).
H.C.F. von Mangoldt. “Beweis der Gleichung \(\sum \limits _{k=1}^{\infty } \frac{\mu (k)}{k}=0\)” . German. In: Berl. Ber. 1897 (1897), pp. 835–852 (cit. on p. 59).
S. Yakubovich. “New summation and transformation formulas of the Poisson, Müntz, Möbius and Voronoi type”. In: Integral Transforms Spec. Funct. 26.10 (2015), pp. 768–795. https://doi.org/10.1080/10652469.2015.1051483 (cit. on p. 61).
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Ramaré, O. (2022). Möbius Inversions. In: Excursions in Multiplicative Number Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-73169-4_6
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