Totient Summatory Function
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, the totient summatory function \Phi(n) is a
summatory function In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
of
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
defined by: :\Phi(n) := \sum_^n \varphi(k), \quad n\in \mathbf It is the number of
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
integer pairs .


Properties

Using
Möbius inversion Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...
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 The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
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 Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold ...
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 Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
, :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 In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...


References

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External links


Totient summatory function

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