Totient summatory function
In number theory, the totient summatory function [math]\displaystyle{ \Phi(n) }[/math] is a summatory function of Euler's totient function defined by:
- [math]\displaystyle{ \Phi(n) := \sum_{k=1}^n \varphi(k), \quad n\in \mathbf{N} }[/math]
It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.
Properties
Using Möbius inversion to the totient function, we obtain
- [math]\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) }[/math]
Φ(n) has the asymptotic expansion
- [math]\displaystyle{ \Phi(n) \sim \frac{1}{2\zeta(2)}n^{2}+O\left( n\log n \right ), }[/math]
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
- [math]\displaystyle{ S(n) := \sum _{k=1}^{n}{\frac {1}{\varphi (k)}} }[/math]
Edmund Landau showed in 1900 that this function has the asymptotic behavior
- [math]\displaystyle{ S(n) \sim A (\gamma+\log n)+ B +O\left(\frac{\log n} n\right) }[/math]
where γ is the Euler–Mascheroni constant,
- [math]\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) }[/math]
and
- [math]\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). }[/math]
The constant A = 1.943596... is sometimes known as Landau's totient constant. The sum [math]\displaystyle{ \textstyle \sum _{k=1}^\infty\frac 1 {k\varphi (k)} }[/math] is convergent and equal to:
- [math]\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 }[/math]
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:
- [math]\displaystyle{ \prod_p \left(1+\frac 1 {p^2(p-1)} \right) = 1.339784\ldots }[/math]
See also
References
- Weisstein, Eric W.. "Totient Summatory Function". http://mathworld.wolfram.com/TotientSummatoryFunction.html.
External links
- Totient summatory function
- Decimal expansion of totient constant product(1 + 1/(p^2*(p-1))), p prime >= 2)
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