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ON SUMS INVOLVING THE EULER TOTIENT FUNCTION
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 24 August 2023, pp. 486-497
- Print publication:
- June 2024
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Let
$\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers 
$n_{1},\ldots ,n_{k}$ and let 
$\phi $ be the Euler totient function. For any real number 
$x>3$ and any integer 
$k\geq 2$, we investigate the asymptotic behaviour of 
$\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $
NOTE ON SUMS INVOLVING THE EULER FUNCTION
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 07 February 2019, pp. 194-200
- Print publication:
- October 2019
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In this note, we provide refined estimates of two sums involving the Euler totient function,
where
$$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$
$[x]$ denotes the integral part of real 
$x$. The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].
A NOTE ON THE DISTRIBUTION FUNCTION OF
$ \varphi (\lowercase {P}-1)/(\lowercase {P}-1)$
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 93 / Issue 1-2 / October 2012
- Published online by Cambridge University Press:
- 16 January 2013, pp. 77-83
- Print publication:
- October 2012
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