Farey sequences
Matematičeskie zametki, Tome 61 (1997) no. 1, pp. 91-113.

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The connection between the distribution of the terms of a Farey sequence and the behavior of the Riemann zeta function is studied.
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S. B. Stechkin. Farey sequences. Matematičeskie zametki, Tome 61 (1997) no. 1, pp. 91-113. http://geodesic.mathdoc.fr/item/MZM_1997_61_1_a8/

[1] Dickson L. E., History of the theory of numbers. V. I. Divisibility and primality, Chelsea Co., N. Y., 1966 | MR | Zbl

[2] Hardy G. H., Wright E. M., An introduction to the theory of numbers, Clarendon Press, Oxford, 1975

[3] Hardy G. H., “Asymptotic formula in combinatory analysis”, Proc. London Math. Soc., 17:25 (1918), 75–115 ; Collected Papers of G. H. Hardy, V. 1, Clarendon Press, Oxford, 1966, 306–339; Collected Papers of S. Ramanujan, Chelsea Co., N. Y., 1962, 276–309 | DOI

[4] Littlewood J. E., “Quelques onséquences de l'hypothèse que la fonction $\zeta(s)$ de Riemann n'a pas de zéros dans le demi-plan $\Re(s)>1/2$”, C. R. Acad. Sci. Paris, 154 (1912), 263–266

[5] Titchmarsh E. C., The theory of the Riemann zeta-function, 2nd edition, Clarendon Press, Oxford, 1986, 412 pp. | MR | Zbl

[6] Franel J., “Les suites de Farey et la problème des nombers premiers”, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Math.-Phisick. Klasse, 1924, 198–201

[7] Landau E., “Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel, premiers”, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Math.-Phisick. Klasse, 1924, 202–206

[8] Landau E., Vorlesungen über Zahlentheorie, V. 2, Chelsea Co., N. Y., 1947

[9] Landau E., “Über die Fareyreihe und die Riemannsche Vermutung”, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Math.-Phisick. Klasse, 1932, 347–352 | Zbl

[10] Huxley M. N., The distribution of prime numbers, Clarendon Press, Oxford, 1972, 128 pp. | MR | Zbl

[11] Mikolás M., “Un théorème d' équivalence et ses applications”, Norske Vid. Selsk. Forh., Frondheim, 22:28 (1950), 128–131 | MR | Zbl

[12] Konyagin S. V., Popov A. Yu., “O skorosti raskhodimosti nekotorykh integralov”, Matem. zametki, 58:2 (1995), 243–255 | MR | Zbl

[13] Landau E., Handbuch der Lehre von der Verteilung der Primzahlen, V. 2, B. G. Teubner, Leipzig, 1909

[14] Ayoub R., An introduction to the analytic theory of numbers, AMS, Providence, 1963, 379 pp. | MR

[15] Ellison W. J., Mendès France M., Les nombers premiers, Hermann, Paris, 1975, 442 pp. | MR | Zbl

[16] Walfisz A., Weyleshe exponentialsummen in der Neueren Zahlen-theorie, Veb Deutscher Verlag der Wissenschaften, Berlin, 1963 | Zbl

[17] Davenport H., “On a generalization of Euler's function $\phi(n)$”, J. London Math. Soc., 7 (1932), 290–296 ; The collected works of Harold Davenport, V. IV, Acad. Press, London, 1977 | DOI | Zbl

[18] Postnikov A. G., Vvedenie v analiticheskuyu teoriyu chisel, Nauka, M., 1971

[19] Sato K., “The Farey sequence and estimation of Franel's sum”, International Conference “Number Theory and Algebraic Methods in Computer Science” (29 June – 2 July, Moscow), M., 1993, 77–78