Asymptotic evaluation of Euler-phi sums of various residue classes
aa r X i v : . [ m a t h . N T ] J un ASYMPTOTIC EVALUATION OF EULER-PHI SUMSOF VARIOUS RESIDUE CLASSES
AMRIK SINGH NIMBRAN
Abstract.
This note contains some asymptotic formulas for the sumsof various residue classes of Euler’s φ -function. Introduction
The phi-function was introduced by Euler in connection with his gener-alization of Fermat’s Theorem. It occurs without the functional notation inhis 1759 paper
Theoremata arithmetica nova methodo demonstrata [6]. In § πD “the multitude of numbersless than D, and which have no common divisor with it” and then providesa table of πD for D = 1 to 100 writing π . Gauss introduced the symbol φ in §
38 of his
Disquitiones Arithmeticae (1801) with φ (1) = 1 . The func-tion φ ( n ) denotes the number of positive integers not exceeding n which arerelatively prime to n. Clearly, for p prime, we have φ ( p ) = p − . As Euler observed (Theorem 3, pp.81–82), if p is a prime, the positiveintegers ≤ p k that are not relatively prime to p k are the p k − multiples of p : p, p, p, . . . , p k − · p. So φ ( p k ) = p k − p k − = p k (1 − p ) = p k − ( p − , and P kj =0 φ ( p j ) = ( p − p + p + · · · + p k − ] = p k . Furthermore, if( a, b ) = 1 , then φ ( a b ) = φ ( a ) φ ( b ) . Thus if m has the prime factorization m = p r p r · · · p r k k , then φ ( m ) = p r − p r − · · · p r k − k ( p − p − · · · ( p k − . And, φ ( m k ) = m k − φ ( m ) . Also, if ( a, b ) = d, then φ ( a b ) = φ ( a ) φ ( b ) dφ ( d ) . As Gauss showed: X d | n φ ( d ) = X φ ( n/d ) = n. The value of φ ( n ) fluctuates as n varies. Since averages sooth out fluc-tuations, it may be fruitful to study the arithmetic mean Φ( n ) n , whereΦ( n ) = P nm =1 φ ( m ) . Date : June 5, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Euler’s φ -function; Residue classes; Sum of prime numbers;Asymptotic summation of φ ( kn ). In 1874, Mertens obtained [3, p.122][11] an asymptotic value for Φ( N ) forlarge N. He employed the function µ ( n ) and proved that G X m =1 φ ( m ) = 12 G X n =1 µ ( n ) ((cid:20) Gn (cid:21) + (cid:20) Gn (cid:21)) = 3 π G + ∆with | ∆ | < G ( ln G + γ + ) + 1 , where γ is Euler’s constant and µ ( n ) isthe M¨obius function defined as µ ( n ) = n = 1 , ( − r if n is product of r distinct prime numbers , n has one or more repeated prime factors . If ( a, b ) = 1 , µ ( a b ) = µ ( a ) µ ( b ) . Further, X d | n µ ( d ) = 0 ( n > . For any positive integer n, we have[1, pp.78–80]: φ ( n ) = X d | n nd µ ( d ) = X d | n d µ (cid:16) nd (cid:17) . It is shown in [8, p.268 Theorem 330][2, pp.61-62] that:Φ( n ) = 3 n π + O ( n ln n ) . (1)To prove (1), we may recall here Euler’s zeta function and identity: ζ ( s ) = ∞ X n =1 n s = Y p − prime (cid:18) − p s (cid:19) − , ℜ ( s ) > . Since for s > , ζ ( s ) = Y p (cid:0) − p − s (cid:1) = Y { µ ( p ) p − s + µ ( p ) p − s + . . . } = ∞ X n =1 µ ( n ) n s and φ ( n ) = n X d | n µ ( d ) d we have:Φ( n ) = n X m =1 φ ( m ) = n X m =1 m X d | m µ ( d ) d = X dd ′ ≤ n d ′ µ ( d ) = n X d =1 µ ( d ) ⌊ nd ⌋ X d ′ =1 d ′ . That is,Φ( n ) = n X d =1 µ ( d ) (cid:26) j nd k (cid:16)j nd k + 1 (cid:17)(cid:27) = 12 n X d =1 µ ( d ) (cid:26) n d + O (cid:16) nd (cid:17)(cid:27) , VALUATION OF EULER-PHI SUMS OF RESIDUE CLASSES 3 leading to Φ( n ) = n n X d =1 µ ( d ) d + O n n X d =1 d ! = n ∞ X d =1 µ ( d ) d − n ∞ X d = n +1 µ ( d ) d + O ( n ln n ) . = n ζ (2) + O n ∞ X d = n +1 d ! + O ( n ln n ) . Or, Φ( n ) = n ζ (2) + O ( n ) + O ( n ln n ) = 3 n π + O ( n ln n ) . Lehmer studied sums of φ ( n ) in [9] and revisited in [10]. I seek here anextension of Lehmer’s formula occurring in [10] by using his argument.2. Asymptotic summation of φ ( pn )Since φ (2 k ) = 2 k − , so: φ (4 m + 2) = φ (2 m + 1); φ (4 m ) = 2 φ (2 m ) . Denoting Φ e ( n ) = X m ≤ n ; m even φ ( m ) and Φ o ( n ) = X m ≤ n ; m odd φ ( m ) , and us-ing the relation:Φ e ( n ) = Φ o ( n/
2) + 2Φ e ( n/
2) = Φ( n/
2) + Φ e ( n/ , Lehmer [10] deduced: Φ e ( n ) = ℓ X λ =1 Φ e ( n/
2) ( ℓ = [ln n/ ln 2]) and then usedthe formula (1) to derive:Φ e ( n ) = (cid:16) nπ (cid:17) + O ( n ln n ); Φ o ( n ) = 2 (cid:16) nπ (cid:17) + O ( n ln n ) . (2)Let Φ r i ( n ) = m X k =1 φ ( kp − i ) , with fixed i = 0 , , , . . . , p − mp − i ) ≤ n. Then Φ r ( n ) = ( p − p − X i =1 Φ r i ( n/p ) + p Φ r ( n/p ) . Hence,Φ r ( n ) = ( p −
1) Φ( n/p ) + Φ r ( n/p ) . Mimicking Lehmer’s proof, we see that for any prime p, Φ r ( n ) = ( p −
1) 3 π − n q X λ =1 p − λ + O ( n log n ) ( q = [ln n/ ln p ])= 3( p − p − π − n + O (cid:18) n Z ∞ q ( p − ) t dt (cid:19) + O ( n log n )= 3 p + 1 π − n + O ( n log n ) . (cid:3) AMRIK SINGH NIMBRAN
The last asymptotic formula implies the following theorem:
Theorem 1.
For any prime p, we have: lim m →∞ m X k =1 φ ( pk )( pm ) = 3( p + 1) π . (3)If the set N is partitioned into p residue classes modulo p, we will have oneclass consisting of composite numbers of the form pm while the remaining p − p : ( p − . The rationale behind the first part of the statement is found in Dirichlet’sfamous theorem relating to primes in arithmetic progressions: every arith-metic progression, with the first member and the difference being coprime,will contain infinitely many primes.
In other words, if k > k, ℓ ) = 1 , then there are infinitely many primes of the form kn + ℓ, where n runs over the positive integers. If k is a prime p, then ℓ is one ofthe numbers 1 , , . . . , p − . Let us recall here the arithmetic function known as the
Mangoldt function which is defined as:Λ( n ) = ( ln p, if n = p m for some prime p and positive integer m, . This function has an important role in elementary proofs of the prime num-ber theorem which states that if π ( n ) denotes the number of primes ≤ n, then π ( n ) ∼ n ln n . We have ([8, pp.253-254]) for n ≥ n ) = X d | n µ (cid:16) nd (cid:17) ln d = X d | n µ ( d ) ln (cid:16) nd (cid:17) = − X d | n µ ( d ) ln d, and X d | n Λ( d ) = ln n. Further [8, p.348][2, p.89], X n ≤ x Λ( n ) n = ln x + O (1) , whence X p ≤ x ln pp = ln x + O (1) . (4)This related result is well-known[2, p.148]: X p ≤ x ; p ≡ ℓ ( mod k ) ln pp = 1 φ ( k ) ln x + O (1) , (5) VALUATION OF EULER-PHI SUMS OF RESIDUE CLASSES 5 where the sum is extended over those primes p ≤ x which are congruent to ℓ (mod k ) . Since ln x → ∞ as x → ∞ this relation implies that there are in-finitely many primes p ≡ ℓ (mod k ) , hence infinitely many in the progression kn + ℓ. Since the principal term on the right hand side in (5) is independentof ℓ, therefore it not only implies Dirichlet’s theorem but it also shows [2,p. 148] that the primes in each of the φ ( k ) reduced residue classes (mod k )make the same contribution to the principal term in (4), that is, the primesare equally distributed among φ ( k ) reduced residue classes ( mod k ) . We thushave a prime number theorem for arithmetic progressions [2, p. 154]: If π ℓ ( x ) counts the number of primes ≤ x in the progression kn + ℓ, then π ℓ ( x ) ∼ π ( x ) φ ( k ) ∼ φ ( k ) x ln x . Hence as m → ∞ , Φ r i ( m ) ∼ Φ r j ( m ) , i, j = 0 . And so, we deduce from(1) and our Theorem 1 the following result:
Theorem 2.
For any prime p, we have for each i = 1 , , , . . . , p − , lim m →∞ m X k =1 φ ( pk − i )( pm ) = 3 p ( p − π ; (6)We will now obtain asymptotic evaluation of the sums of residue classesmodulo p for the φ -function.Since φ (4 m −
2) = φ (2 m − φ (4 m ) = 2 φ (2 m ) and as n → ∞ , Φ(2 n −
1) = n X m =1 φ (2 m −
1) = 2 Φ(2 n ) = 2 n X m =1 φ (2 m ) , so we have:lim n →∞ Φ(4 n − n ) = lim n →∞ Φ(4 n )(4 n ) = 12 π . Further, as n → ∞ ; Φ(2 n −
1) = Φ(4 n −
3) + Φ(4 n −
1) = 2 Φ(2 n ) =2 Φ(4 n −
2) + 2 Φ(4 n ) and the two forms 4 k − , k − n →∞ Φ(4 n − n ) = lim n →∞ Φ(4 n − n ) = 1 π . Again, lim n →∞ Φ(6 n − n ) + lim n →∞ Φ(6 n − n ) + lim n →∞ Φ(6 n )(6 n ) = 1 π and lim n →∞ Φ(6 n − n ) + lim n →∞ Φ(6 n − n ) + lim n →∞ Φ(6 n − n ) = 2 π . Further lim n →∞ Φ(6 n − n ) = lim n →∞ Φ(6 n − n ) = 32 lim n →∞ Φ(6 n )(6 n ) and lim n →∞ Φ(6 n − n ) = lim n →∞ Φ(6 n − n ) = 32 lim n →∞ Φ(6 n − n ) . Still further, lim n →∞ Φ(3(2 n − n ) = 2 lim n →∞ Φ(3(2 n ))(6 n ) . So we deduce these results:lim n →∞ Φ(6 n − n ) = lim n →∞ Φ(6 n − n ) = 38 π . lim n →∞ Φ(6 n − n ) = 12 π ; lim n →∞ Φ(6 n )(6 n ) = 14 π . lim n →∞ Φ(6 n − n ) = lim n →∞ Φ(6 n − n ) = 34 π . In fact, we have following the general theorem based on two facts: (i)the sum of all odd residue classes equals twice the sum of all even classes,and (ii) the ratio of residue classes modulo p containing primes to the classhaving only composite numbers is pp − . Theorem 3.
For an odd prime p, lim n →∞ Φ(2 pn − (2 p − pn ) = lim n →∞ Φ(2 pn − pn ) =lim n →∞ Φ(2 pn − (2 p − pn ) = lim n →∞ Φ(2 pn − pn ) = . . . lim n →∞ Φ(2 pn − ( p + 1))(2 pn ) = lim n →∞ Φ(2 pn − ( p − pn ) = p ( p − π ;lim n →∞ Φ(2 pn )(2 pn ) = 1( p + 1) π . And lim n →∞ Φ(2 pn − (2 p − pn ) = lim n →∞ Φ(2 pn − pn ) =lim n →∞ Φ(2 pn − (2 p − pn ) = lim n →∞ Φ(2 pn − pn ) = . . . lim n →∞ Φ(2 pn − ( p + 2))(2 pn ) = lim n →∞ Φ(2 pn − ( p − pn ) = 2 p ( p − π ;lim n →∞ Φ(2 pn − p )(2 pn ) = 2( p + 1) π . VALUATION OF EULER-PHI SUMS OF RESIDUE CLASSES 7
Remark. If m < p, then m cannot divide p. Also, p cannot divide m and m − simultaneously; it may not divide either. So gcd ( p, m ) = 1 or p andgcd ( p, m −
1) = p or . Hence, φ ( p (2 m )) = ( p − φ (2 m ) or p φ (2 m ); and φ ( p (2 m − p φ (2 m − or ( p − φ (2 m − depending on m. Lehmerproved that lim n →∞ Φ(2 n − n ) = 2 . Hence, lim n →∞ P nm =1 φ ( p (2 m − P nm =1 φ ( p (2 m )) = 2 . Acknowledgement:
The author is thankful to Prof Paul Levrie for hishelpful comments and the anonymous referee for his suggestions which madethe presentation concise.
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