Totient summatory function

Applying Möbius inversion to the totient function yields :\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), where \mu(n) is the Möbius function. has the asymptotic expansion :\Phi(n) \sim \frac{1}{2\zeta(2)}n^{2}+O\left( n\log n \right ) = \frac{3}{\pi^2}n^2+O\left( n\log n \right), where is the Riemann zeta function evaluated at 2, which is \frac{\pi^2}{6}. == Reciprocal totient summatory function ==

The summatory function of the reciprocal of the totient is :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 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\in\mathbb{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\in\mathbb{P}}\left(\frac {\log p}{p^2-p+1}\right). The constant is sometimes known as '''Landau's totient constant'''. The sum \textstyle \sum _{k=1}^\infty 1 / (k \; \varphi (k)) converges to :\sum _{k=1}^\infty \frac 1 {k\varphi (k)} = \zeta(2) \prod_{p\in\mathbb{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 the totient summatory constant, and its value is :\prod_{p\in\mathbb{P}} \left(1+\frac 1 {p^2(p-1)} \right) = 1.339784\ldots. == See also ==