Summary
Let k and l be integers such that 2⩽k ⩽l. Let Sk and S ′l lbe two subsets of positive integers with Sk⊒Qk and S′l ⊒ Ql, where Qk denotes the set of k-free integers. Further suppose that the characteristic functions of Sk and S ′l l are multiplicative. Let T(n) denote the number of representations of n in the form n=a+b, where a ε Sk and b∈ S ′l l. In this paper we establish an asymptotic formula for T(n), when n is sufficiently large; and deduce several known asymptotic formulae as particular cases.
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References
E. Brintzer,Über (k, r)-Zahlen, Monatsh. Math.,80 (1975), pp. 31–35.
L. Carlitz,On a problem in additive arithmetic, Quart. J. Math. (Oxford),3 (1932), pp. 273–290.
E. Cohen,Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged),22 (1961), pp. 223–233.
E. Cohen,Arithmetical Notes III, certain equally distributed sets of integers, Pacific J. Math.,12 (1962), pp. 77–84.
E. Cohen,Arithmetical Notes XIII, A sequal to note IV, Ele. Math.,18 (1963), pp. 8–11.
E. Cohen,The number of representations of an integer as a sum of two square-free numbers, Duke Math. J.,32 (1965), pp. 181–185.
E. Cohen -R. L. Robinson,On the distribution of the k-free integers in residue classes, Acta Arith.,8 (1962–63), pp. 283–293; Errata, ibid.,10 (1964–65), p. 443; Correction, ibid.,16 (1969–70), p. 439.
T. Estermann,On the representation of a number as a sum of two numbers not divisible by k-th powers, J. London Math. Soc.,6 (1931), pp. 37–40.
C. J. A. Evelyn -E. H. Linfoot,On a problem in the additive theory of numbers (second paper), J. Reine. Angew. Math.,164 (1931), pp. 131–140.
G. E. Hardy,The distribution of (k, r)-free integers in residue classes, Notices Amer. Math. Soc.,23 (1976), p. A.53.
G. H. Hardy -E. M. Wright,An introduction to the theory of numbers, 4th ed., Calender press, Oxford, 1960.
A.Page,An asymptotic formula in the theory of numbers, J. London Math. Soc. (1932), pp. 24–27.
C. Pomerance -D. Suryanarayana,On a problem of Evelyn-Linfoot and Page in additive number theory, Publicationes Mathematicae, Debrecen,26 (1979), pp. 237–244.
K. Prachar,Über die kleinste quadratfreie Zahl einer arithmetischen Reihe, Monatsh. Math.,62 (1958), pp. 173–176.
V.Sita Ramaiah - D.Suryanarayana,The number of representations of an integer as the sum of a prime and an element in a set, Bollettino U.M.I, (to appear).
M. V. Subbarao -V. C. Harris,A new generalization of Ramanujan's sum, J. London Math. Soc.,41 (1966), pp. 595–604.
M. V. Subbarao -Y. K. Feng,On the distribution of generalized k-free integers in residue classes, Duke Math. J.,38 (1971), pp. 741–748.
D. Suryanarayana,Semi-k-free integers, Ele. Math.,26 (1971), pp. 39–40.
B. M. Wilson,Proofs of some formulae enuntiated by Ramanujan, Proc. London Math. Soc., (2)24 (1922), pp. 235–255.
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Suryanarayana, D., Sita Ramaiah, V. Generalization of a problem of Evelyn-Linfoot and Page in additive number theory. Annali di Matematica pura ed applicata 126, 1–17 (1980). https://doi.org/10.1007/BF01762499
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DOI: https://doi.org/10.1007/BF01762499