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# Totient summatory function

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${\displaystyle \Phi (n):=\sum _{k=1}^{n}\varphi (k),\quad n\geq 1,\,}$

where ${\displaystyle \scriptstyle \varphi (n)\,}$ is Euler's totient function.

 20204     ${\displaystyle \Phi (n)\,}$
0

## Asymptotic behavior

${\displaystyle \Phi (n)={\frac {3n^{2}}{\pi ^{2}}}+O(n\log n).\,}$

More precisely,

${\displaystyle \Phi (n)={\frac {3n^{2}}{\pi ^{2}}}+O((n\log n)^{2/3}(\log \log n)^{4/3}).\,}$

(A. Walfisz 1963)[1]Benoit Cloitre, Feb 02 2003

Is there a relation with the asymptotic behavior of the summatory quadratfrei function or is it a coincidence? (Cf. Talk:Totient summatory function#Asymptotic behavior of totient summatory function.)

## Sequences

The totient summatory function (partial sums of Euler's totient function) (Cf. A092249 and A002088 for ${\displaystyle \scriptstyle n\,\geq \,1\,}$) gives (all values for ${\displaystyle \scriptstyle n\,\geq \,2\,}$ are even)

{1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, ...}

## Notes

1. A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin 1963.