Some sums involving Farey fractions I

Shigeru KANEMITSU; R. SITA RAMA CHANDRA RAO; A. SIVA RAMA SARMA

doi:10.2969/jmsj/03410125

Shigeru KANEMITSU, R. SITA RAMA CHANDRA RAO, A. SIVA RAMA SARMA

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ジャーナルフリー

1982 年 34 巻 1 号 p. 125-142

DOI https://doi.org/10.2969/jmsj/03410125

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Published: 1982 Received: 1980/01/01 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) T.M. Apostol, Modular functions and Dirichlet's series in number theory, Springer-Verlag, Berlin, 1976.
2) W.E. Briggs, S. Chowla, A.J. Kempner and W.E. Mientka, On some infinite series, Scripta Math., 21 (1955), 28-30.
3) H. Davenport, On a generalization of Euler's function φ(n), J. London Math. Soc., 7 (1932), 290-296.
4) L.E. Dickson, History of the theory of numbers, Vol. 1, Chelsea Publishing Company, New York 1952.
5) H. Gupta, An identity, Res. Bull. Panjab Univ. (N.S.), 15 (1964/65), 347-349.
6) R.R. Hall, A note on Farey series, J. London Math. Soc., (2), 2, (1970), 139-148.
7) R.J. Hans and V.C. Dumir, An interesting identity, Res. Bull. Panjab Univ. (N.S.), 15 (1964/65), 353-356.
8) G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon Press, Oxford, 1965.
9) S. Kanemitsu, On some sums involving Farey fractions, Math. J. Okayama Univ., 20 (1978), 101-113.
10) N.M. Korobov, Estimates of trigonometrical sums and their applications, Uspehi Mat. Nauk, 13 (1958), No. 4(82), 185-192, (Russian).
11) J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arith., 15 (1968/69), 181-187.
12) M. Mikolás, Farey series and their connection with the prime number problem I, Acta Univ. Szeged. Sect Sci. Math., 13 (1949), 93-117.
13) M. Mikolás, Farey series and their connection with the prime number problem II, Acta Univ. Szeged. Sect. Sci. Math., 14 (1951/52), 5-21.
14) T. Mitsui, Theory of numbers (An introduction to analytic number theory), Shibundo, Tokyo, 1970, (Japanese).
15) S.S. Pillai and S.D. Chowla, On the error terms in some asymptotic formulae in the theory of numbers (I), J. London Math. Soc., 5 (1930), 95-101.
16) H. Rademacher, Topics in analytic number theory, Springer-Verlag, Berlin, 1973.
17) A. Saltykov, On Euler's function, Vestnik Moskov. Univ. Ser. I Mat. Meh., No. 6, 6 (1960), 34-50, (Russian).
18) R. Sita Rama Chandra Rao and A. Siva Rama Sarma, Some identities involving the Riemann's zeta-function, Indian J. Pure Appl. Math., 10 (1979), 602-607.
19) D. Suryanarayana, On the average order of the function E(x)=∑<n_??_x>φ(n)-3x²/π² (II), J. Indian Math. Soc., 42 (1978), 179-195.
20) A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.
21) G.T. Williams, A new method of evaluating ζ(2n), Amer. Math. Monthly, 60 (1953), 19-25.

Right : [1] T. M. Apostol, Modular functions and Dirichlet's series in number theory, Springer-Verlag, Berlin, 1976.
[2] W. E. Briggs, S. Chowla, A. J. Kempner and W. E. Mientka, On some infinite series, Scripta Math., 21 (1955), 28-30.
[3] H. Davenport, On a generalization of Euler's function φ(n), J. London Math. Soc., 7 (1932), 290-296.
[4] L. E. Dickson, History of the theory of numbers, Vol. 1, Chelsea Publishing Company, New York 1952.
[5] H. Gupta, An identity, Res. Bull. Panjab Univ. (N. S.), 15 (1964/65), 347-349.
[6] R. R. Hall, A note on Farey series, J. London Math. Soc., (2), 2, (1970), 139-148.
[7] R. J. Hans and V. C. Dumir, An interesting identity, Res. Bull. Panjab Univ. (N. S.), 15 (1964/65), 353-356.
[8] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon Press, Oxford, 1965.
[9] S. Kanemitsu, On some sums involving Farey fractions, Math. J. Okayama Univ., 20 (1978), 101-113.
[10] N. M. Korobov, Estimates of trigonometrical sums and their applications, Uspehi Mat. Nauk, 13 (1958), No. 4 (82), 185-192, (Russian).
[11] J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arith., 15 (1968/69), 181-187.
[12] M. Mikolás, Farey series and their connection with the prime number problem I, Acta Univ. Szeged. Sect Sci. Math., 13 (1949), 93-117.
[13] M. Mikolás, Farey series and their connection with the prime number problem II, Acta Univ. Szeged. Sect. Sci. Math., 14 (1951/52), 5-21.
[14] T. Mitsui, Theory of numbers (An introduction to analytic number theory), Shibundo, Tokyo, 1970, (Japanese).
[15] S. S. Pillai and S. D. Chowla, On the error terms in some asymptotic formulae in the theory of numbers (I), J. London Math. Soc., 5 (1930), 95-101.
[16] H. Rademacher, Topics in analytic number theory, Springer-Verlag, Berlin, 1973.
[17] A. Saltykov, On Euler's function, Vestnik Moskov. Univ. Ser. I Mat. Meh., No. 6, 6 (1960), 34-50, (Russian).
[18] R. Sita Rama Chandra Rao and A. Siva Rama Sarma, Some identities involving the Riemann's zeta-function, Indian J. Pure Appl. Math., 10 (1979), 602-607.
[19] D. Suryanarayana, On the average order of the function E(x)=∑_n≤xφ(n)-3x²/π² (II), J. Indian Math. Soc., 42 (1978), 179-195.
[20] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.
[21] G. T. Williams, A new method of evaluating ζ (2n), Amer. Math. Monthly, 60 (1953), 19-25.

訂正日: 2006/10/20 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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