totient, summatory, function, number, theory, totient, summatory, function, displaystyle, summatory, function, euler, totient, function, defined, displaystyle, varphi, quad, mathbf, number, coprime, integer, pairs, contents, properties, summatory, reciprocal, . In number theory the totient summatory function F n displaystyle Phi n is a summatory function of Euler s totient function defined by F n k 1 n f k n N displaystyle Phi n sum k 1 n varphi k quad n in mathbf N It is the number of coprime integer pairs p q 1 p q n Contents 1 Properties 2 The summatory of reciprocal totient function 3 See also 4 References 5 External linksProperties editUsing Mobius inversion to the totient function we obtain F n k 1 n k d k m d d 1 2 k 1 n m k n k 1 n k displaystyle Phi n sum k 1 n k sum d mid k frac mu d d frac 1 2 sum k 1 n mu k left lfloor frac n k right rfloor left 1 left lfloor frac n k right rfloor right nbsp F n has the asymptotic expansion F n 1 2 z 2 n 2 O n log n displaystyle Phi n sim frac 1 2 zeta 2 n 2 O left n log n right nbsp where z 2 is the Riemann zeta function for the value 2 F n is the number of coprime integer pairs p q 1 p q n The summatory of reciprocal totient function editThe summatory of reciprocal totient function is defined as S n k 1 n 1 f k displaystyle S n sum k 1 n frac 1 varphi k nbsp Edmund Landau showed in 1900 that this function has the asymptotic behavior S n A g log n B O log n n displaystyle S n sim A gamma log n B O left frac log n n right nbsp where g is the Euler Mascheroni constant A k 1 m k 2 k f k z 2 z 3 z 6 p 1 1 p p 1 displaystyle A sum k 1 infty frac mu k 2 k varphi k frac zeta 2 zeta 3 zeta 6 prod p left 1 frac 1 p p 1 right nbsp and B k 1 m k 2 log k k f k A p log p p 2 p 1 displaystyle B sum k 1 infty frac mu k 2 log k k varphi k A prod p left frac log p p 2 p 1 right nbsp The constant A 1 943596 is sometimes known as Landau s totient constant The sum k 1 1 k f k displaystyle textstyle sum k 1 infty frac 1 k varphi k nbsp is convergent and equal to k 1 1 k f k z 2 p 1 1 p 2 p 1 2 20386 displaystyle sum k 1 infty frac 1 k varphi k zeta 2 prod p left 1 frac 1 p 2 p 1 right 2 20386 ldots nbsp In this case the product over the primes in the right side is a constant known as totient summatory constant 1 and its value is p 1 1 p 2 p 1 1 339784 displaystyle prod p left 1 frac 1 p 2 p 1 right 1 339784 ldots nbsp See also editArithmetic functionReferences edit OEIS A065483 Weisstein Eric W Totient Summatory Function MathWorld External links editTotient summatory function Decimal expansion of totient constant product 1 1 p 2 p 1 p prime gt 2 nbsp This mathematics related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Totient summatory function amp oldid 1096402460, wikipedia, wiki, book, books, library,