Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

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1 EulerPhi Notatios Traditioal ame Euler totiet fuctio Traditioal otatio Φ Mathematica StadardForm otatio EulerPhi Primary defiitio Φ gcd,k, ; For oegative iteger, the Euler totiet fuctio Φ is the umber of positive itegers less tha ad relatively prime to Φ Φ ; Example: There are exist oly 4 positive itegers less tha 0 ad relatively prime to 0; they are, 3, 7, ad 9 (because, for example, gcd 0, 9 but gcd 0, 8 2 ); so the Euler totiet fuctio Φ 0 4. By defiitio, Φ 0 4. Specific values Specialized values Φ p p p ; p Values at fixed poits Φ 0 0

2 Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ 20 8

3 Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ 40 6

4 Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Geeral characteristics Domai ad aalyticity Φ is a oaalytical fuctio which is defied oly for iteger Φ Symmetries ad periodicities Parity Φ is a eve fuctio.

5 Φ Φ Symmetry No symmetry Periodicity No periodicity Series represetatios Other series represetatios Φ ; gcd k, Φ gcd,k, ; p Φ d j j d j p d j k 0 exp 2 Π j k d j ; p d j divisors Φ d j Μ d j d j ; d j divisors Μ d j Φ ; d j divisors d j d j Product represetatios m Φ ; factors p,,, p m, m p k p k Idetities Fuctioal idetities Φ r c Φ c ; c Μ d j Φ d j p k k p k k d j

6 Φ k Φ k ; k Φ m Φ gcd, m Φ Φ m ; m gcd, m Μ d j 2 Φ ; d j divisors Φ d j d j Summatio Fiite summatio l Φ k 2 l Φ d j ; d j divisors d j Φ d j Σ 0 d j d j Σ ; d j divisors Φ d j d j Σ k d j Σ k ; d j divisors k Ifiite summatio Φ Ζ s ; Re s s Ζ s k 2 Φ k f f x x ; 0 i case x f x C 0,. Asymptotic fiite summatio Φ k Π 2 2 O log 4 3 log 3 log ;

7 Φ k k x O log 3 log 3 log ; Π 2 Φ k Ζ 3 35 Ζ 3 log 2 Π 4 2 Π 4 Μ k 2 log k k Φ k O log ; log Φ k log a 2 log O k 3 log 2 ; a 2 log p k p k p k Asymptotic ifiite summatio Ζ 2 Ζ 3 Θ x Φ k x R x ; x R x x ε log x log log x 2 ε 0 Ζ 6 Operatios Limit operatio lim Φ k 3 2 Π 2 lim k f 2 Φ k 6 Π 2 0 t f t t Φ 2 lim mi Φ 2 lim max Φ, Φ 3 Φ Φ, Φ 3 Φ 2,, Φ Φ,, Φ Φ lim max 2 Φ Φ 2, 3 Φ Φ 3,, Φ Φ Φ 3 lim log max 3 log log 3, Φ 4 4 log log 4,, Φ log log Φ lim mi log log k,,

8 8 Represetatios through equivalet fuctios With related fuctios m s pi,q j i j Φ sg ; factors p,,, p m, m factors k q,,, q s, s Φ p Λ p ; p p 2 x Π x k Φ k k ; x Σ 0 Σ Φ d j ; d j divisors d j d j Σ Φ d j Σ 0 d d j j ; d j divisors Φ gcd m,d j, Μ Μ Φ d j d j d j d j d j gcd m, d j Μ d j ; d j divisors m Iequalities Φ ; Φ 2 3 ; 42 Φ log log log log ; log x Φ ; 2 x x 6 log x m Φ ; p k k p k k p k p k k m m Φ log 2 2 log ; 3

9 Φ m Φ Φ m Φ m Φ m Φ Φ m 2 Φ Φ Σ 0 ; Φ log log Σ 0 ; Φ Σ ; Φ Π ; Φ Σ ; 2 Φ Φ 2 ; 2 Σ mi Σ Φ 2 Σ Φ 3,, max Φ Φ 2 Φ Φ 3,, Zeros Φ 0 0 Other idetities Cogruece properties Φ mod 2 0 ; 2 Theorems Fermat-Euler theorem a Φ m mod m ; gcd a, m

10 0 Geeralized Fermat-Euler theorem a k Φ b mod b a k mod b ; gcd a k, b gcd a k, b k. The solutio of a liear cogruece The solutio of a x b mod m 0 is give by x b d a d Φ m d mod m d ; d gcd a, m b d. Number of differet values of Φ k for k The umber V x of differet values occurrig i the list Φ, Φ 2,, Φ x behaves asymptotically as where Farey fractios The umber of irreducible fractios betwee 0 ad with deomiator is Φ. Geerators of cyclic groups Φ is the umber of geerators of a cyclic group of order. Carmichael cojecture For every it is possible to fid a m m, such that Φ Φ m. Number of ecklaces The umber of uique fixed ecklaces of legth l which ca be made out of b differet beads is l l d l Φ d b d. Probability that two radomly choose positive itegers are relatively prime The probability p 2 that two radomly choose positive itegers ad 2, 2 x are relatively prime gcd, 2 is give by x p 2 x 2 2 Φ k x Ζ 2 6 Π. 2 Prime roots All itegers of the form, 2, 4, p k, 2 p k where 2 p, k, have Φ Φ differet primitive roots. History

11 L. Euler (760, 763) C. F. Gauss (80) itroduced the symbol Φ J. J. Sylvester (879) itroduced the ame "totiet fuctio" E. Cesaro (888) evaluated asymptotics for cumulative sums E. Ladau (900)

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