From Wikipedia, the free encyclopedia
Arithmetic function
In number theory, the totient summatory function
is a summatory function of Euler's totient function defined by:
![{\displaystyle \Phi (n):=\sum _{k=1}^{n}\varphi (k),\quad n\in \mathbf {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1736c46e948614697fedc5261127ea7e23bca903)
It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.
Properties[edit]
Using Möbius inversion to the totient function, we obtain
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ad49ff1ab00915a2097dcf5d967188065305a5)
Φ(n) has the asymptotic expansion
![{\displaystyle \Phi (n)\sim {\frac {1}{2\zeta (2)}}n^{2}+O\left(n\log n\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c19332ecb2b639e2a4b0b6ac6b51bde13a3adf8a)
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[edit]
The summatory of reciprocal totient function is defined as
![{\displaystyle S(n):=\sum _{k=1}^{n}{\frac {1}{\varphi (k)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f007bd37c8bfed7a1636a13b7b237e8fce843e)
Edmund Landau showed in 1900 that this function has the asymptotic behavior
![{\displaystyle S(n)\sim A(\gamma +\log n)+B+O\left({\frac {\log n}{n}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b2f542dd9d8283c89521d70637572f4b25fcd1)
where γ is the Euler–Mascheroni constant,
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf030849ea32676b81c7ffe43cb231fdf9ae698)
and
![{\displaystyle 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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1354b27d78e4109ee6415497d3d6b433459b891a)
The constant A = 1.943596... is sometimes known as Landau's totient constant. The sum
is convergent and equal to:
![{\displaystyle \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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b43abc6f003e86598041009649a88778a0cf699)
In this case, the product over the primes in the right side is a constant known as totient summatory constant,[1] and its value is:
![{\displaystyle \prod _{p}\left(1+{\frac {1}{p^{2}(p-1)}}\right)=1.339784\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3058d84eade0aa5d4d44f364afa829a9e6aba2af)
See also[edit]
References[edit]
External links[edit]