Geodesic
Some Generalizations of an Identity of Subhankulov
Canadian mathematical bulletin, Tome 20 (1977) no. 4, pp. 489-494

Voir la notice de l'article provenant de la source Cambridge University Press

In 1957, M. A. Subhankulov established the following identity where ; μ is the Môbius function and J 2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(nr) = ∑d| (n, r)dμ(r/d), we rewrite the above identity using C(n, r).In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.
DOI : 10.4153/CMB-1977-073-6
Mots-clés : 10A20, 10A99, Möbius function, Jordan totient function, generalized Ramanujan's sum, k-free integers, Riemann zeta function 
Suryanarayana, D.; Walker, David T. Some Generalizations of an Identity of Subhankulov. Canadian mathematical bulletin, Tome 20 (1977) no. 4, pp. 489-494. doi: 10.4153/CMB-1977-073-6
@article{10_4153_CMB_1977_073_6,
     author = {Suryanarayana, D. and Walker, David T.},
     title = {Some {Generalizations} of an {Identity} of {Subhankulov}},
     journal = {Canadian mathematical bulletin},
     pages = {489--494},
     year = {1977},
     volume = {20},
     number = {4},
     doi = {10.4153/CMB-1977-073-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-073-6/}
}
TY  - JOUR
AU  - Suryanarayana, D.
AU  - Walker, David T.
TI  - Some Generalizations of an Identity of Subhankulov
JO  - Canadian mathematical bulletin
PY  - 1977
SP  - 489
EP  - 494
VL  - 20
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-073-6/
DO  - 10.4153/CMB-1977-073-6
ID  - 10_4153_CMB_1977_073_6
ER  - 
%0 Journal Article
%A Suryanarayana, D.
%A Walker, David T.
%T Some Generalizations of an Identity of Subhankulov
%J Canadian mathematical bulletin
%D 1977
%P 489-494
%V 20
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-073-6/
%R 10.4153/CMB-1977-073-6
%F 10_4153_CMB_1977_073_6

[1] 1. Cohen, E., An extension of Ramanujan's sum, Duke Math. J.,' 16 (1949), 85-90. Google Scholar

[2] 2. Cohen, E., Some totient functions, Duke Math. J., 23 (1956), 515-522. Google Scholar

[3] 3. Cohen, E., Generalizations of the Euler φ-function, Scripta Math., 23 (1957), 157-161. Google Scholar

[4] 4. Cohen, E., Trigonometric sums in elementary number theory, Amer. Math. Monthly, 66 (1959), 105-117. Google Scholar

[5] 5. Cohen, E., A class of arithmetical functions in several variables with applications to congruences, Trans. Amer. Math. Soc, 96 (1960), 355-381. Google Scholar

[6] 6. Cohen, E., A trigonometric sum, Math. Student, 28 (1960), 29-32. Google Scholar

[7] 7. Dickson, L.E., History of the theory of numbers, Vol. I, Chelsea Publishing Company, reprinted, 1952. Google Scholar

[8] 8. Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers, Fourth edition, Oxford University Press, 1960. Google Scholar

[9] 9. Klee, V.L., A generalization ofEulefs function, Amer. Math. Monthly, 55 (1948), 358-359. Google Scholar

[10] 10. Subhankulov, M.A., Some asymptotic formulas in additive theory of numbers (Russian), Scientific Journal of the Tadzhik University, Vol. X, No. 4, (1957), 15-22. Google Scholar

[11] 11. Subhankulov, M.A. and Muhatarov, S.N., Representations of a number as a sum of two square-free numbers (Russian), Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1960, No. 4, 3-10. Google Scholar

[12] 12. Sugunamma, M., Eckford Cohen's generalizations of Ramanujan's trigonometrical sum C(n, r), Duke Math. J., 27 (1960), 323-330. Google Scholar

Cité par Sources :