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Uniform O-estimates of certain error functions connected with k-free integers

Published online by Cambridge University Press:  09 April 2009

D. Suryanarayana
Affiliation:
Department of MathematicsAndhra University Waltair, India
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Let k be a fixed integer ≧ 2. A positive integer m is called k-free if m is not divisible by the k'th power of any integer > 1. Let qk(m) be the characteristic function of the set of k-free integers; that is, qk(m) = 1 or 0 according as m is k-free or not. It can be easily shown that where μ(n) is the Mobius function. Let x ≧ 1 denote a real variable and n be a positive integer. Let Qk(x, n) and be the number and the sum of the reciprocals of the k-free integers ≦ x which are prime to n respectively.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Bergmann, G. M., ‘Solution of the problem 5091’, Amer. Math. Monthly 71 (1964), 334335.Google Scholar
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[4]Dickson, L. E., History of the Theory of numbers, Vol. I (Chelsea reprint, New York, 1952).Google Scholar
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[6]Robinson, R. L., ‘An estimate for the enumerative functions of certain sets of integers’, Proc. Amer. Math. Soc. 17 (1966), 232237.CrossRefGoogle Scholar
[7]Suryanarayana, D., ‘The number of k-ary divisors of an integer’, Monatsh. Math. 72 (1968), 445450.CrossRefGoogle Scholar
[8]Suryanarayana, D., ‘The greatest divisor of n which is prime to k’, Maths Student 36 (1968), 171181.Google Scholar
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