Totient Summatory Constant

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

:$\Phi(n) := \sum_^n \varphi(k), \quad n\in \mathbf$

It is the number of coprime integer pairs .

Properties

Using Möbius inversion to the totient function, we obtain

:$\Phi(n) = \sum_^n k\sum _ \frac = \frac \sum _^n \mu(k) \left\lfloor \frac \right\rfloor \left(1 + \left\lfloor \frac \right\rfloor \right)$

has the asymptotic expansion

:$\Phi(n) \sim \fracn^+O\left( n\log n \right ),$

where is the Riemann zeta function for the value 2.

is the number of coprime integer pairs .

The summatory of reciprocal totient function

The summatory of reciprocal totient function is defined as

:$S(n) := \sum _^$

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

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

where is the Euler–Mascheroni constant,

:$A = \sum_^\infty \frac = \frac = \prod_p \left(1+\frac 1 \right)$

and

:$B = \sum_^ \frac = A \, \prod _\left(\frac \right).$

The constant is sometimes known as Landau's totient constant. The sum $\textstyle \sum _^\infty\frac 1$ is convergent and equal to:

:$\sum _^\infty \frac 1 = \zeta(2) \prod_p \left(1 + \frac 1 \right) =2.20386\ldots$

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

:$\prod_p \left(1+\frac 1 \right) = 1.339784\ldots$

See also

* Arithmetic function

References

*

External links

Totient summatory function


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

Arithmetic functions

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