On the order of some error functions related to $k$-free integers
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- by V. S. Joshi
- Proc. Amer. Math. Soc. 35 (1972), 325-332
- DOI: https://doi.org/10.1090/S0002-9939-1972-0337839-X
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Erratum: Proc. Amer. Math. Soc. 51 (1975), 251-252.
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 \[ \Delta {’_k}(x) - x{\Delta _k}(x) = O({x^{1 + 1/2k + \varepsilon }})\] and \[ \frac {1}{x}\int _1^x {{\Delta _k}(t)dt = O({x^{1/2k + \varepsilon }}),} \] for arbitrary $\varepsilon > 0$.References
- E. Hille, Analysis. Vol. I, Blaisdell, Waltham, Mass., 1964. Theorem B.2.3, p. 333.
- D. Suryanarayana and R. Sitaramachandra Rao, On the order of the error function of the $k$-free integers, Proc. Amer. Math. Soc. 28 (1971), 53–58. MR 271044, DOI 10.1090/S0002-9939-1971-0271044-X
- E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485 —, The theory of functions, Clarendon Press, Oxford, 1939, pp. 44-45.
- A. M. Vaidya, On the order of the error function of the square-free numbers, Proc. Nat. Inst. Sci. India Part A 32 (1966), 196–201. MR 249378
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 325-332
- MSC: Primary 10H25
- DOI: https://doi.org/10.1090/S0002-9939-1972-0337839-X
- MathSciNet review: 0337839