Totient summatory function

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


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

It is the number of coprime integer pairs ${p, q}, 1 ≤ p ≤ q ≤ n$.

Properties
Using Möbius inversion to the totient function, we obtain


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

$Φ(n)$ has the asymptotic expansion


 * $$\Phi(n) \sim \frac{1}{2\zeta(2)}n^{2}+O\left( n\log n \right ),$$

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

$Φ(n)$ is the number of coprime integer pairs ${p, q}, 1 ≤ p ≤ q ≤ n$.

The summatory of reciprocal totient function
The summatory of reciprocal totient function is defined as


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

Edmund Landau showed in 1900 that this function has the asymptotic 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


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

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:


 * $$\sum _{k=1}^\infty \frac 1 {k\varphi (k)} = \zeta(2) \prod_p \left(1 + \frac 1 {p^2(p-1)}\right) =2.20386\ldots $$

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


 * $$\prod_p \left(1+\frac 1 {p^2(p-1)} \right) = 1.339784\ldots$$