Totient summatory function

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

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

It is the number of ordered pairs of coprime integers , where .

The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, ... . Values for powers of 10 are 1, 32, 3044, 304192, 30397486, 3039650754, ... .

Properties

Applying Möbius inversion to the totient function yields

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

has the asymptotic expansion

:$\Phi(n) \sim \fracn^+O\left( n\log n \right ) = \fracn^2+O\left( n\log n \right),$

where is the Riemann zeta function evaluated at 2, which is $\frac$.

Reciprocal totient summatory function

The summatory function of the reciprocal of the totient is

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

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,

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

and

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

The constant is sometimes known as Landau's totient constant. The sum $\textstyle \sum _^\infty 1 / (k \; \varphi (k))$ converges to

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

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

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

See also

*Arithmetic function

References

*

External links

* OEIS Totient summatory function

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

Arithmetic functions

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