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Sur Quelques Series de Lambert et de Dirichlet

  • Jacques Touchard (a1)

Extract

Nous nous occupons dans ce travail de séries dont voici le type le plus simple

et de leurs analogues lorsqu'on remplace xn par n-s et yn par n-s . Faute d'une autre désignation, nous avons cru pouvoir appeler ces séries respectivement séries de Lambert et séries de Dirichlet. Une série plus générale que la précédente est

où les ak sont des constantes et où g(x) est un polynôme dont les coefficients sont des symboles de Jacobi (1, pp. 132-40; 4 pp 361-9).

possédant la période 2ω. Les expressions

ont des propriétés multiplicatives analogues à celles des sommes de Ramanujan (6) d'où l'on peut déduire deux propositions concernant les racines de g(x).

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References

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1. Bachmann, P., Zahlentheorie, Teil 1.
2. Bachmann, P. Zahlentheorie, Teil 2.
3. Brunel, G., Monographie de la fonction gamma.
4. Cahen, E., Théorie des nombres, T. 2.
5. Euler, L., Oeuvres, Series Prima Vol. X.
6. Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers (2nd éd.; Oxford, 1954).
7. Landau, E., Primzahlen, vol. I.
8. Landau, E. Primzahlen, vol. II.
9. Lindelöf, E., Le Calcul des résidus.
10. Nielsen, N., Traité élémentaire des nombres de Bernoulli.
11. Schlömilch, O., Compendium der Höheren analysis, vol. II.
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Sur Quelques Series de Lambert et de Dirichlet

  • Jacques Touchard (a1)

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