Totient summatory function

In number theory, the totient summatory function $\Phi (n)$ is a summatory function of Euler's totient function defined by:

$\Phi (n):=\sum _{k=1}^{n}\,\varphi (k),\quad n\in \mathbf {N}$

Properties

Using Möbius inversion to the totient function, we obtain

$\Phi (n)=\sum _{k=1}^{n}\,k\sum _{d\mid k}{\frac {\mu (d)}{d}}={\frac {1}{2}}\left(1+\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor ^{2}\right)$

Φ(n) has the asyntotic expansion

$\Phi (x)\sim {\frac {3}{2\zeta (2)}}x^{2}+O\left(x\log x\right).$

where ζ(2) is the Riemann zeta function for the value 2.

See also

• Arithmetic function

References

• Weisstein, Eric W. "Totient Summatory Function". MathWorld.

External links

• Decimal expansion of totient constant product(1 + 1/(p^2*(p-1))), p prime >= 2)