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

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

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

The summatory of reciprocal totient function

The summatory of reciprocal totient funcion is defined as

$S(n):=\sum _{k=1}^{n}{\frac {1}{\varphi (k)}}$

Edmund Landau showed in 1900 that this function has the asymtotic behavior

$S(n)\sim A\,(\gamma +\log n)+B+O\left({\frac {\log n}{n}}\right)$

where γ is the Euler-Mascheroni constant,

$A=\sum _{k=1}^{\infty }{\frac {\mu (k)^{2}}{k\,\varphi (k)}}={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}=\prod _{p}\left(1+{\frac {1}{p(p-1)}}\right)$

and

$B=\sum _{k=1}^{\infty }{\frac {\mu (k)^{2}}{k\,\varphi (k)}}=A\,\prod _{p}\left({\frac {\log p}{p^{2}-p+1)}}\right).$

The constant A=1.943596... is sometimes known as Landau's totient constant. The sum $\textstyle \sum _{k=1}^{\infty }{\frac {1}{k\,\varphi (k)}}$ is convergent and equal to:

$\sum _{k=1}^{\infty }{\frac {1}{k\,\varphi (k)}}=\zeta (2)\,\prod _{p}\left(1+{\frac {1}{p^{2}(p-1)}}\right)=2.20386...$

In this case, the product over the primes in the right side is a constant known as Totient summatory constant, and its value is:

$\prod _{p}\left(1+{\frac {1}{p^{2}(p-1)}}\right)=1.339784...$

See also

• Arithmetic function

References

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

External links

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