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In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the totient summatory function \Phi(n) is a summatory function 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 \mathbb. It is the number of ordered pairs of
coprime In number theory, 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 equiv ...
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 Moebius, Mœbius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Friedrich Möbius (art historian) (1928–2024), German art historian and architectural historian * Theodor ...
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 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 and its analytic c ...
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 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 Leopo ...
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 called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
, :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 In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ...


References

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External links

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