number theory, the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} is a summatory function of Euler's totient function defined by: Φ ( n )...

In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the...

summation function for large x. A classical example of this phenomenon is given by the divisor summatory function, the summation function of d(n), the...

noncototient; however, the totient summatory function over the first thirteen integers is 58. On the other hand, the Euler totient of 58 is the second perfect...

Euler's totient summatory function Φ ( n ) {\displaystyle \Phi (n)} over the first 57 integers. In decimal, multiples of one thousand are totient values...

is the totient summatory function. Neither of the methods mentioned previously leads to practical algorithms to calculate the Mertens function. Using...

{\displaystyle p^{5}} where p {\displaystyle p} is prime. 32 is the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} over the first 10 integers, and...

related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. The sum of positive divisors function σz(n)...

formulas more generally for the related summatory functions over so-termed factorial moments of the function ω ( n ) {\displaystyle \omega (n)} . A known...

/ ln 2: 83  It is conjectured that the Mertens function, or summatory function of the Möbius function, satisfies lim sup n → ∞ | M ( x ) | x = + ∞ , {\displaystyle...

{\displaystyle \mu (n)} is the Moebius function. Another unique Dirichlet series identity generates the summatory function of some arithmetic f evaluated at...

1 = Id {\displaystyle \phi *1={\text{Id}}} , proved under Euler's totient function ϕ = Id ∗ μ {\displaystyle \phi ={\text{Id}}*\mu } , by Möbius inversion...

(f)=q^{2n}(1-q^{-1}).} Divisor summatory function Normal order of an arithmetic function Extremal orders of an arithmetic function Divisor sum identities Hardy...

|F_{n}|=1+\sum _{m=1}^{n}\varphi (m)=1+\Phi (n),} where Φ(n) is the summatory totient. We also have : | F n | = 1 2 ( 3 + ∑ d = 1 n μ ( d ) ⌊ n d ⌋ 2 )...

\prod _{p}\left(1-{\frac {1}{p^{2}(p+1)}}\right)=0.881513...} The totient summatory constant OEIS: A065483: ∏ p ( 1 + 1 p 2 ( p − 1 ) ) = 1.339784......

Euler's totient function: gcd ( a , b ) = ∑ k | a  and  k | b φ ( k ) . {\displaystyle \gcd(a,b)=\sum _{k|a{\text{ and }}k|b}\varphi (k).} GCD Summatory function...

average order summatory functions over an arithmetic function f ( n ) {\displaystyle f(n)} defined as a divisor sum of another arithmetic function g ( n ) {\displaystyle...

number. 108, the second Achilles number. 255, 28 − 1, the smallest perfect totient number that is neither a power of three nor thrice a prime; it is also...