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Proceedings of the American Mathematical Society

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On $ \Delta (x,\,n)=\phi(x,\,n)$ $ -x\phi(n)/n$


Author: D. Suryanarayana
Journal: Proc. Amer. Math. Soc. 44 (1974), 17-21
MSC: Primary 10A20
DOI: https://doi.org/10.1090/S0002-9939-1974-0332636-5
MathSciNet review: 0332636
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Abstract: Let $ \Delta (x,n) = \varphi (x,n) - x\varphi (n)/n$, where $ \varphi (x,n)$ denotes the number of positive integers $ \leqq x$ and prime to $ n,\varphi (n) = \varphi (n,n)$. In this paper, lower and upper bounds for $ \Delta (x,n)$, which hold for all values of $ x \geqq 1$ and $ n \geqq 2$, are established.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0332636-5
Keywords: Legendre and Euler totient functions, Dedekind's $ \psi $-function, the number of square-free divisors of $ n$
Article copyright: © Copyright 1974 American Mathematical Society

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