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On the order of some error functions related to $ k$-free integers


Author: V. S. Joshi
Journal: Proc. Amer. Math. Soc. 35 (1972), 325-332
MSC: Primary 10H25
DOI: https://doi.org/10.1090/S0002-9939-1972-0337839-X
Erratum: Proc. Amer. Math. Soc. 51 (1975), 251-252.
MathSciNet review: 0337839
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Abstract: Let $ {\Delta _k}(x)$ and $ \Delta {'_k}(x)$ be the error functions in the asymptotic formulae for the number and the sum of k-free integers not exceeding x. We prove that on the assumption of Riemann hypothesis, we have

$\displaystyle \Delta {'_k}(x) - x{\Delta _k}(x) = O({x^{1 + 1/2k + \varepsilon }})$

and

$\displaystyle \frac{1}{x}\int_1^x {{\Delta _k}(t)dt = O({x^{1/2k + \varepsilon }}),} $

for arbitrary $ \varepsilon > 0$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0337839-X
Keywords: k-free integers, Riemann hypothesis
Article copyright: © Copyright 1972 American Mathematical Society

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