THE RELATION AMONG EULER S PHI FUNCTION, TAU FUNCTION, AND SIGMA FUNCTION

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1 International Journal of Pure and Applied Mathematics Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v118i3.15 PAijpam.eu THE RELATION AMONG EULER S PHI FUNCTION, TAU FUNCTION, AND SIGMA FUNCTION Apisit Pakapongpun Department of Mathematics Faculty of Science Burapha University Chonburi, 0131, THAILAND Abstract: The purpose of this study was to investigate the relationship among Euler s phi function, tau function and sigma function. Using knowledge of number theory, the relationship of these functions and provide the proofs was found. AMS Subject Classification: 11A5 Key Words: Euler s Phi function, Tau function, Sigma function 1. Introduction Definition 1.1. An arithmetic function is a function that is defined for all positive integers. Definition 1.. An arithmetic function f is called multiplicative if and only if f(mn) = f(m)f(n) where m and n are relatively primepositive integers. Definition 1.3. Euler s phi function denoted by φ is defined by setting φ(n)equal tothenumberoftheinteger lessthanorequaltonthatarerelatively prime to n. Received: Revised: Published: April 15, 018 c 018 Academic Publications, Ltd. url:

2 676 A. Pakapongpun Theorem 1.4. The following statements are true 1. φ is a multiplicative function.. Let n = p a 1 1 pa pa k k is theprime power factorization into distinct primes of the positive integer n. Then φ(n) = n k (1 1.) p i i=1 Definition 1.5. Tau function or the number of divisors function, denoted by τ is defined by setting τ(n) equal to the number of positive divisors of n. Theorem 1.6. The following statements are true 1. τ is a multiplicative function.. Let n = p a 1 1 pa pa k k is theprime power factorization into distinct primes of the positive integer n. Then τ(n) = k (a i +1). i=1 Definition 1.7. Sigma function or the sum of divisors function, denoted by σ is defined by setting σ(n) equal to the sum of all the positive divisors of n. Theorem 1.8. The following statements are true 1. σ is a multiplicative function.. Let n = p a 1 1 pa pa k k is theprime power factorization into distinct primes of the positive integer n. Then σ(n) = k i=1 p a i+1 i 1 p i 1. Definition 1.1, 1., 1.3, 1.5, 1.7 and theorem 1.4, 1.6, 1.8 are taken from [1], [].

3 THE RELATION AMONG EULER S PHI FUNCTION Main Results Lemma.1. Let a is a non negative integer and p is a positive prime number. If n = p a, then 1 k n,(k,n)=1 k = n φ(n). Proof. Let A be the sum of positive integers less than or equal to p a and B be the sum of positive integers r less than or equal to p a and (r,p a ) 1. So 1 k n,(k,n)=1 k = A B = pa (p a +1) p(pa 1 )(p a 1 +1) = pa (pa +1+p a 1 1) = pa (pa p a 1 ) = n φ(n). Theorem.. If k and n are positive integers, then 1 k n,(k,n)=1 k = n φ(n). Proof. Case I, if n is a prime. Then 1 k n,(k,n)=1 k = (n 1) = n(n 1) = n φ(n). Case II, if n is not a prime and n = p a 1 1 pa pam m such that p 1,p,,p m are distinct primes and a 1,a,,a m are positive integers, then k = m 1( )( ) 1 k n,(k,n)=1 1 k 1 p a 1 1,(k 1,p a 1 1 )=1 k 1 ( 1 k m p am m,(k m,p am m )=1 k m ) = m 1( p a 1 1 (pa 1 1 pa ) )( p a (pa 1 k p a,(k,p a )=1 k pa 1 ) )

4 678 A. Pakapongpun (p am m (pam m p am 1 m ) ) = m 1 m (pa 1 1 pa pam m )(p a 1 1 pa )(p a pa 1 ) (p am m p am 1 m ) = 1 (n)φ(n). Theorem.3. Let a and n are positive integers. If n = a and a+1 1 is a prime number, then σ(σ(n)) = n = τ(n). Proof. Since hence and σ(n) = σ( a ) = a+1 1, σ(σ(n)) = σ( a+1 1) = a+1 = n τ(n) = τ( a ) = a+1. σ(σ(n)) = n = a = a+1 = τ(n). Theorem.4. If p is a prime number, then σ(p) = φ(p)+τ(p). p φ(p) τ(p) σ(p) φ(p) + τ(p) Table 1: some p of σ(p) = φ(p)+τ(p).

5 THE RELATION AMONG EULER S PHI FUNCTION Proof. Let p is a prime number. Then σ(p) = p+1, φ(p) = p 1 and τ(p) =, hence φ(p)+τ(p) = p 1+ = p+1 = σ(p). σ(p) = φ(p)+τ(p) where p is a prime number. Theorem.5. If n = p and p is an odd prime number, then σ(n) = n+φ(n)+τ(n). p n = p φ(n) τ(n) σ(n) n+φ(n)+τ(n) Table : some n of σ(n) = n+φ(n)+τ(n). Proof. Let n = p and p is an odd prime number. Then σ(n) = σ(p) = σ()σ(p) = 3(p+1) = 3p+3, φ(n) = φ(p) = φ()φ(p) = p 1 and τ(n) = τ(p) = τ()τ(p) = () = 4, hence n+φ(n)+τ(n) = 3p+3. σ(n) = n+φ(n)+τ(n) where n = p and p is an odd prime number.

6 680 A. Pakapongpun Theorem.6. If n = 3p and p is a prime number not equal to 3, then σ(n) = (φ(n)+τ(n)). p n = 3p φ(n) τ(n) σ(n) (φ(n)+τ(n)) Table 3: some n of σ(n) = (φ(n)+τ(n)). Proof. Let n = 3p and p is a prime number not equal to 3. Then σ(n) = σ(3p) = σ(3)σ(p) = 4(p+1) = 4p+4, φ(n) = φ(3p) = φ(3)φ(p) = p and τ(n) = τ(3p) = τ(3)τ(p) = () = 4, hence (φ(n)+τ(n)) = (p +4) = 4p+4. σ(n) = (φ(n)+τ(n)) where n = 3p and p is a prime number not equal to 3. Theorem.7. If n = k and k is a non negative integer, then σ(n) = n 1. Proof. Let n = k and k is a non negative integer. Then σ(n) = σ( k ) = k σ(n) = n 1. = n 1.

7 THE RELATION AMONG EULER S PHI FUNCTION k n = k σ(n) n Table 4: some n of σ(n) = n 1. Theorem.8. If n = 1 or n = p is a prime number, then σ(n)+φ(n) = n. n n φ(n) σ(n) φ(n) + σ Table 5: some n of σ(n)+φ(n) = n. Proof. Where n = 1 is obvious. If n = p is a prime number, then σ(n) = σ(p) = p+1 and φ(n) = φ(p) = p 1. σ(n)+φ(n) = p = n where n = 1 or n = p is a prime number. Theorem.9. If p is a prime number, then φ(p) = p (τ(p)) +3.

8 68 A. Pakapongpun p τ(p) (τ(p)) φ(p) p (τ(p)) Table 6: some p of φ(p) = p (τ(p)) +3. Proof. Let p is a prime number. Then φ(p) = p 1 and τ(p) =, hence p (τ(p)) +3 = p +3 = p 1. φ(p) = p (τ(p)) +3 where p is a prime number. Theorem.10. If n = p and p is an odd prime number, then φ(n) = n 1. n p n = p φ(n) Table 7: some n of φ(n) = n 1. Proof. Let n = p and p is an odd prime number. Then φ(n) = φ(p) = p 1 = p 1 = n 1.

9 THE RELATION AMONG EULER S PHI FUNCTION φ(n) = n 1 where n = p and p is an odd prime number. Theorem.11. If n = 4p and p is an odd prime number, then φ(n) = n. n p n = 4p φ(n) Table 8: some n of φ(n) = n. Proof. Let n = 4p and p is an odd prime number. Then φ(n) = φ(4p) = (p 1) = p = 4p = n. φ(n) = n where n = 4p and p is an odd prime number. Acknowledgements I would like to thank you to Stuart for his comments and suggestions on this work and this work is supported by Faculty of Science, Burapha University, Thailand. References [1] Kenneth H. Rosen. (015). Elementary Number Theory ( ed.). India: Pearson India Education Services Pvt. Ltd.

10 684 A. Pakapongpun [] Tom M. Apostol. Introduction to analytic number theory. Sprinnger-Verlag, New York, Undergraduate Texts in Mathematics.

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