Average Orders of the Euler Phi Function, The Dedekind Psi Function, The Sum of Divisors Function, And The Largest Integer Function
aa r X i v : . [ m a t h . G M ] J a n Average Order of the Euler Phi Function And The LargestInteger Function
N. A. Carella
Abstract : Let x ≥ x ] = x − { x } be the largest integer function,and let ϕ ( n ) be the Euler totient function. The result P n ≤ x ϕ ([ x/n ]) = (6 /π ) x log x + O (cid:0) x (log x ) / (log log x ) / (cid:1) was proved very recently. This note presents a short elemen-tary proof, and sharpen the error term to P n ≤ x ϕ ([ x/n ]) = (6 /π ) x log x + O ( x ). Inaddition, the first proofs of the asymptotics formulas for the finite sums P n ≤ x ψ ([ x/n ]) =(15 /π ) x log x + O ( x ), and P n ≤ x σ ([ x/n ]) = ( π / x log x + O ( x ) are also evaluated here. Contents
Some new analytic techniques for evaluating the fractional finite sums P n ≤ x f ([ x/n ]) forslow growing functions f ( n ) ≪ n ε , were recently introduced in [2], and for faster growingfunctions in the more recent literature as [4]. In this note, the standard analytic techniquesoriginally developed for evaluating the average orders P n ≤ x f ( n ) of arithmetic functionsare modified to handle the fractional finite sums P n ≤ x f ([ x/n ]) for fast growing multi-plicative functions, approximately f ( n ) ≫ n (log n ) b , where b ∈ Z . The modified standardtechniques are simpler, more efficient and produce very short proofs. As demonstrations,the fractional finite sum P n ≤ x ϕ ([ x/n ]) of the Euler phi function ϕ in Theorem 2.1, the January 8, 2021
MSC2020 : Primary 11N37, Secondary 11N05.
Keywords : Arithmetic function; Euler phi function; Dedekind psi function, Sum of divisors function;Average orders. uler Phi Function and Largest Integer Function P n ≤ x ψ ([ x/n ]) of the Dedekind psi function ψ in Theorem 4.1, andthe fractional finite sum P n ≤ x σ ([ x/n ]) of the sum of divisors function σ in Theorem6.1, are evaluated here. The three functions ϕ ( n ) ≤ ψ ( n ) ≤ σ ( n ) share many similari-ties such as multiplicative structures, rates of growths, et cetera, and have similar proofs.Theorem 2.1 has a very short proof, and sharpen the error term of a very recent result P n ≤ x ϕ ([ x/n ]) = (6 /π ) x log x + O (cid:0) x (log x ) / (log log x ) / (cid:1) proved in [4] using a verycomplicated and lengthy proof. Further, Theorem 4.1, and Theorem 6.1 are new resultsin the literature. The first result deals with the Euler totient function ϕ ( n ) = n P d | n µ ( d ) /d . It is multi-plicative and satisfies the growth condition ϕ ( n ) ≫ n/ log log n . A very short proof for P n ≤ x ϕ ([ x/n ]) is produced here. It is a modified version of the standard proof for theaverage order P n ≤ x ϕ ( n ) = (3 /π ) x + O ( x log x ), which appears in [1, Theorem 3.7], andsimilar references. Theorem 2.1. If x ≥ is a large number, then, X n ≤ x ϕ (cid:16)h xn i(cid:17) = 6 π x log x + O ( x ) . (1) Proof.
Use the identity ϕ ( n ) = P d | n µ ( d ) /d to rewrite the finite sum, and switch the orderof summation: X n ≤ x ϕ (cid:16)h xn i(cid:17) = X n ≤ x h xn i X d | [ x/n ] µ ( d ) d (2)= X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] h xn i . Expanding the bracket yields X n ≤ x ϕ (cid:16)h xn i(cid:17) = X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] (cid:16) xn − n xn o(cid:17) (3)= x X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n − X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n xn o = S ( x ) − S ( x ) . Therefore, the difference of the subsum S ( x ) computed in Lemma 3.2 and the subsum S ( x ) computed in Lemma 3.3 complete the verification. (cid:4) It is easy to verify that the subsums S ( x ) and S ( x ) imply the omega result X n ≤ x ϕ (cid:16)h xn i(cid:17) − π x log x = Ω ± ( x ) (4)or a better result. uler Phi Function and Largest Integer Function The detailed and elementary proofs of the preliminary results required in the proof ofTheorem 2.1 concerning the Euler phi function ϕ ( n ) = P d | n µ ( d ) d are recorded in thissection. Theorem 3.1. ([2, Theorem 2.1])
For x ≥ , ax log x ≤ X n ≤ x ϕ (cid:16)h xn i(cid:17) ≤ bx log x, (5) where a > and b > are constants. Lemma 3.1.
Let x ≥ be a large number, and let ≤ d, n ≤ x be integers. Then, d X ≤ a ≤ d − e i πa [ x/n ] /d = (cid:26) if d | [ x/n ] , if d ∤ [ x/n ] , (6) Proof.
The two cases d | [ x/n ] and d ∤ [ x/n ] are easily handled with the basic exponentialsum X ≤ k ≤ q − e i πkm/q = (cid:26) q if q | m, q ∤ m, (7)where m = 0, and q ≥ (cid:4) Lemma 3.2.
Let x ≥ be a large number. Then, x X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n = 6 π x log x + O ( x ) . (8) Proof.
Apply Lemma 3.1 to remove the congruence on the inner sum index, and break itup into two subsums. Specifically, S ( x ) = x X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n (9)= x X d ≤ x µ ( d ) d X n ≤ x n · d X ≤ a ≤ d − e i πa [ x/n ] /d = x X d ≤ x µ ( d ) d X n ≤ x n + x X d ≤ x µ ( d ) d X n ≤ x n X
1) (10)= x X d ≤ x µ ( d ) d X n ≤ x n . uler Phi Function and Largest Integer Function d ∤ n and Case 2. d | n . Case 1. If d ∤ n , substitute the standard asymptotic formula for the harmonic sum P n ≤ x /n , and the asymptotic formula for the finite sum of Mobius function P n ≤ x µ ( n ) /n = O (cid:16) e − c √ log x (cid:17) , see [1, Theorem 4.14], then, S ( x ) = x X d ≤ x µ ( d ) d X n ≤ x n (11)= x X d ≤ x µ ( d ) d (cid:18) log x + γ + O (cid:18) x (cid:19)(cid:19) = O (cid:16) ( x log x ) e − c √ log x (cid:17) , where c > S ( x ) = o ( x log x ) contradicts Theorem 3.1. Hence, d | n . Case 2. To evaluate this case, let dm = n ≤ x , and substitute the standard asymptoticfor the harmonic sum P m ≤ x/d /m to obtain the following. S ( x ) = x X d ≤ x µ ( d ) d X m ≤ x/d m (12)= x X d ≤ x µ ( d ) d (cid:18) log (cid:16) xd (cid:17) + γ + O (cid:18) dx (cid:19)(cid:19) = x log x X d ≤ x µ ( d ) d − x X d ≤ x µ ( d ) log dd + γx X d ≤ x µ ( d ) d + O X d ≤ x d = 6 π x log x + O ( x ) , where γ is Euler constant, and the partial sums X n ≤ x µ ( n ) n = 1 ζ (2) + O (cid:18) x (cid:19) , (13)where 1 /ζ (2) = 6 /π , see [1, Theorem 3.13], and its ‘derivative’ − X n ≤ x µ ( n ) log nn = ζ ′ (2) ζ (2) + O (cid:18) log xx (cid:19) , (14)are convergent series, bounded by constants. (cid:4) Lemma 3.3. Let x ≥ be a large number. Then, X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n xn o = O ( x ) . (15) Proof. Apply Lemma 3.1 to remove the congruence on the inner sum index, and break it uler Phi Function and Largest Integer Function S ( x ) = X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n xn o (16)= X d ≤ x µ ( d ) d X n ≤ x n xn o · d X ≤ a ≤ d − e i πa [ x/n ] /d = X d ≤ x µ ( d ) d X n ≤ x n xn o + X d ≤ x µ ( d ) d X n ≤ x n xn o X
1) (17)= X d ≤ x µ ( d ) d X n ≤ x n xn o . Since d | n , see (11), and (12), substitute dm = n ≤ x , and complete the estimate: S ( x ) = X d ≤ x µ ( d ) d X m ≤ x/d n xdm o (18) ≤ x X d ≤ x d = O ( x ) , where the partial sum P n ≤ x /n is bounded by a constant, see [1, Theorem 3.2]. (cid:4) The second result deals with the Dedekind function ψ ( n ) = n P d | n µ ( d ) /d . It is multi-plicative and satisfies the growth condition ψ ( n ) ≫ n . The first asymptotic formula forthe fractional finite sum of the Dedekind function is given below. Theorem 4.1. If x ≥ is a large number, then, X n ≤ x ψ (cid:16)h xn i(cid:17) = 15 π x log x + O ( x ) . (19) Proof. Use the identity ψ ( n ) = n P d | n µ ( d ) /d to rewrite the finite sum, and switch theorder of summation: X n ≤ x ψ (cid:16)h xn i(cid:17) = X n ≤ x h xn i X d | [ x/n ] µ ( d ) d (20)= X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] h xn i . uler Phi Function and Largest Integer Function X n ≤ x ϕ (cid:16)h xn i(cid:17) = X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] (cid:16) xn − n xn o(cid:17) (21)= x X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n − X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n xn o = S ( x ) − S ( x ) . Therefore, the difference of the subsum S ( x ) computed in Lemma 5.1 and the subsum S ( x ) computed in Lemma 5.2 complete the verification. (cid:4) It is easy to verify that the subsums S ( x ) and S ( x ) imply the omega result X n ≤ x ψ (cid:16)h xn i(cid:17) − π x log x = Ω ± ( x ) (22)or a better result. A sketch of the standard proof for the average order X n ≤ x ψ ( n ) = 15 π x + O ( x log x ) , (23)appears in [1, Exercise 13, p. 71]. The detailed and elementary proofs of the preliminary results required in the proof ofTheorem 4.1 concerning the Dedekind psi function ψ ( n ) = P d | n µ ( d ) d are recorded inthis section. Theorem 5.1. For a large number x ≥ , ax log x ≤ X n ≤ x ψ (cid:16)h xn i(cid:17) ≤ bx log x, (24) where a > and b > are constants.Proof. Since ϕ ( n ) ≤ ψ ( n ), the lower bound ax log x ≤ P n ≤ x ψ ([ x/n ]) follows from Theo-rem 3.1. The upper bound is computed via the identity X n ≤ x ψ (cid:16)h xn i(cid:17) = X n ≤ x ψ ( n ) (cid:18)h xn i − (cid:20) xn + 1 (cid:21)(cid:19) (25)= x X n ≤ x ψ ( n ) n ( n + 1) + X n ≤ x ψ ( n ) (cid:18) − n xn o + (cid:26) xn + 1 (cid:27)(cid:19) . The difference of consecutive fractional parts has the upper bound (cid:12)(cid:12)(cid:12)(cid:12) − n xn o + (cid:26) xn + 1 (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − n xn o + (cid:26) xn − xn ( n + 1) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:26) n | x, cxn ( n +1) if n ∤ x. (26) uler Phi Function and Largest Integer Function x ≥ X n | x ψ ( n ) ≤ X n | x n log log n ≤ x X n | x n ≤ x (log log x ) , (27)(recall that ψ ( n ) ≤ n log log n ). Hence, partial summation, and the result in (23) lead to X n ≤ x ψ (cid:16)h xn i(cid:17) ≤ b x X n ≤ x ψ ( n ) n ( n + 1) + X n | x ψ ( n ) (28) ≤ bx log x, where b , b, c > (cid:4) Lemma 5.1. Let x ≥ be a large number. Then, x X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n = 15 π x log x + O ( x ) . (29) Proof. Apply Lemma 3.1 to remove the congruence on the inner sum index, and break itup into two subsums. Specifically, S ( x ) = x X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n (30)= x X d ≤ x µ ( d ) d X n ≤ x n · d X ≤ a ≤ d − e i πa [ x/n ] /d = x X d ≤ x µ ( d ) d X n ≤ x n + x X d ≤ x µ ( d ) d X n ≤ x n X
1) (31)= x X d ≤ x µ ( d ) d X n ≤ x n . There are two possible evaluations of (31): Case 1. d ∤ n and Case 2. d | n . Case 1. If d ∤ n , substitute the standard asymptotic formula for the harmonic sum P n ≤ x /n , then, S ( x ) = x X d ≤ x µ ( d ) d X n ≤ x n (32)= x X d ≤ x µ ( d ) d (cid:18) log x + γ + O (cid:18) x (cid:19)(cid:19) = 15 π x log x + O ( x ) . uler Phi Function and Largest Integer Function S ( x ) ≫ x log x contradicts Theorem 5.1. Hence, d | n . Case 2. To evaluate this case, let dm = n ≤ x , and substitute the standard asymptoticfor the harmonic sum P m ≤ x/d /m to obtain the following. S ( x ) = x X d ≤ x µ ( d ) d X m ≤ x/d m (33)= x X d ≤ x µ ( d ) d (cid:18) log (cid:16) xd (cid:17) + γ + O (cid:18) dx (cid:19)(cid:19) = x log x X d ≤ x µ ( d ) d − x X d ≤ x µ ( d ) log dd + γx X d ≤ x µ ( d ) d + O X d ≤ x d = 15 π x log x + O ( x ) , where γ is Euler constant, and the partial sums X n ≤ x µ ( n ) n = ζ (2) ζ (4) + O (cid:18) x (cid:19) , (34)where ζ (2) /ζ (4) = 15 /π , and its ‘derivative’ − X n ≤ x µ ( n ) log nn = − c + O (cid:18) log xx (cid:19) , (35)where c > (cid:4) Lemma 5.2. Let x ≥ be a large number. Then, X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n xn o = O ( x ) . (36) Proof. Apply Lemma 3.1 to remove the congruence on the inner sum index, and break itup into two subsums. Specifically, S ( x ) = X d ≤ x µ ( d ) d X n ≤ xd | [ x/n ] n xn o (37)= X d ≤ x µ ( d ) d X n ≤ x n xn o · d X ≤ a ≤ d − e i πa [ x/n ] /d = X d ≤ x µ ( d ) d X n ≤ x n xn o + X d ≤ x µ ( d ) d X n ≤ x n xn o X
1) (38)= X d ≤ x µ ( d ) d X n ≤ x n xn o . uler Phi Function and Largest Integer Function d | n , see (32), and (33), substitute dm = n ≤ x , and complete the estimate: S ( x ) = X d ≤ x µ ( d ) d X m ≤ x/d n xdm o (39) ≤ x X d ≤ x d = O ( x ) , where the partial sum P n ≤ x /n is bounded by a constant, see [1, Theorem 3.2]. (cid:4) The third result deals with the sum of divisors function σ ( n ) = n P d | n /d . It is multi-plicative and satisfies the growth condition σ ( n ) ≫ n . The first asymptotic formula forthe fractional sum of divisor function is given below. Theorem 6.1. If x ≥ is a large number, then, X n ≤ x σ (cid:16)h xn i(cid:17) = π x log x + O ( x ) . (40) Proof. Use the identity σ ( n ) = n P d | n /d to rewrite the finite sum, and switch the orderof summation: X n ≤ x σ (cid:16)h xn i(cid:17) = X n ≤ x h xn i X d | [ x/n ] d (41)= X d ≤ x d X n ≤ xd | [ x/n ] h xn i . Expanding the bracket yields X n ≤ x σ (cid:16)h xn i(cid:17) = X d ≤ x d X n ≤ xd | [ x/n ] (cid:16) xn − n xn o(cid:17) (42)= x X d ≤ x d X n ≤ xd | [ x/n ] n − X d ≤ x d X n ≤ xd | [ x/n ] n xn o = S ( x ) − S ( x ) . Therefore, the difference of the subsum S ( x ) computed in Lemma 7.1 and the subsum S ( x ) computed in Lemma 7.2 complete the verification. (cid:4) It is easy to verify that the subsums S ( x ) and S ( x ) imply the omega result X n ≤ x σ (cid:16)h xn i(cid:17) − π x log x = Ω ± ( x ) (43)or a better result. The standard proof for the average order X n ≤ x σ ( n ) = π x + O ( x log x ) , (44)appears in [1, Theorem 3.4]. uler Phi Function and Largest Integer Function The detailed and elementary proofs of the preliminary results required in the proof ofTheorem 6.1 concerning the sum of divisor function σ ( n ) = P d | n d are recorded in thissection. Theorem 7.1. For a large number x ≥ , ax log x ≤ X n ≤ x σ (cid:16)h xn i(cid:17) ≤ bx log x, (45) where a > and b > are constants.Proof. Since ϕ ( n ) ≤ σ ( n ), the lower bound ax log x ≤ P n ≤ x σ ([ x/n ]) follows from Theo-rem 3.1. The upper bound is computed via the identity X n ≤ x σ (cid:16)h xn i(cid:17) = X n ≤ x σ ( n ) (cid:18)h xn i − (cid:20) xn + 1 (cid:21)(cid:19) (46)= x X n ≤ x σ ( n ) n ( n + 1) + X n ≤ x σ ( n ) (cid:18) − n xn o + (cid:26) xn + 1 (cid:27)(cid:19) . The difference of consecutive fractional parts has the upper bound (cid:12)(cid:12)(cid:12)(cid:12) − n xn o + (cid:26) xn + 1 (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − n xn o + (cid:26) xn − xn ( n + 1) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:26) n | x, cxn ( n +1) if n ∤ x. (47)Moreover, if x ≥ X n | x σ ( n ) ≤ X n | x n log log n ≤ x X n | x n ≤ x (log log x ) , (48)(recall that σ ( n ) ≤ n log log n ). Hence, partial summation, and the result in (23) lead to X n ≤ x σ (cid:16)h xn i(cid:17) ≤ b x X n ≤ x σ ( n ) n ( n + 1) + X n | x σ ( n ) (49) ≤ bx log x, where b , b, c > (cid:4) Lemma 7.1. Let x ≥ be a large number. Then, x X d ≤ x d X n ≤ xd | [ x/n ] n = π x log x + O ( x ) . (50) Proof. Apply Lemma 3.1 to remove the congruence on the inner sum index, and break itup into two subsums. Specifically, S ( x ) = x X d ≤ x d X n ≤ xd | [ x/n ] n (51)= x X d ≤ x d X n ≤ x n · d X ≤ a ≤ d − e i πa [ x/n ] /d = x X d ≤ x d X n ≤ x n + x X d ≤ x d X n ≤ x n X
1) (52)= x X d ≤ x d X n ≤ x n . There are two possible evaluations of (52): Case 1. d ∤ n and Case 2. d | n . Case 1. If d ∤ n , substitute the standard asymptotic formula for the harmonic sum P n ≤ x /n , then, S ( x ) = x X d ≤ x d X n ≤ x n (53)= x X d ≤ x d (cid:18) log x + γ + O (cid:18) x (cid:19)(cid:19) = π x log x + O ( x ) . But, S ( x ) ≫ x log x contradicts Theorem 7.1. Hence, d | n . Case 2. To evaluate this case, let dm = n ≤ x , and substitute the standard asymptoticfor the harmonic sum P m ≤ x/d /m to obtain the following. S ( x ) = x X d ≤ x d X m ≤ x/d m (54)= x X d ≤ x d (cid:18) log (cid:16) xd (cid:17) + γ + O (cid:18) dx (cid:19)(cid:19) = x log x X d ≤ x d − x X d ≤ x log dd + γx X d ≤ x d + O X d ≤ x d = π x log x + O ( x ) , where γ is Euler constant, and the partial sums X n ≤ x n = π O (cid:18) x (cid:19) , (55)and its ‘derivative’ − X n ≤ x log nn = − ζ ′ (2) + O (cid:18) log xx (cid:19) , (56)are convergent series, bounded by constants. The constant − ζ ′ (2) = 0 . . . . , archivedin OEIS A073002, has a rather complicated expression, see [3, Eq. 25.6.15]. (cid:4) uler Phi Function and Largest Integer Function Lemma 7.2. Let x ≥ be a large number. Then, X d ≤ x d X n ≤ xd | [ x/n ] n xn o = O ( x ) . (57) Proof. Apply Lemma 3.1 to remove the congruence on the inner sum index, and break itup into two subsums. Specifically, S ( x ) = X d ≤ x d X n ≤ xd | [ x/n ] n xn o (58)= X d ≤ x d X n ≤ x n xn o · d X ≤ a ≤ d − e i πa [ x/n ] /d = X d ≤ x d X n ≤ x n xn o + X d ≤ x d X n ≤ x n xn o X
1) (59)= X d ≤ x d X n ≤ x n xn o . Since d | n , see (53), and (54), substitute dm = n ≤ x , and complete the estimate: S ( x ) = X d ≤ x d X m ≤ x/d n xdm o (60) ≤ x X d ≤ x d = O ( x ) , where the partial sum P n ≤ x /n is bounded by a constant, see [1, Theorem 3.2]. (cid:4) Small numerical tables were generated by an online computer algebra system, the rangeof numbers x ≤ is limited by the wi-fi bandwidth. The error terms are defined by E ( x ) = X n ≤ x ϕ (cid:16)h xn i(cid:17) − π x log x, (61)and E ( x ) = X n ≤ x σ (cid:16)h xn i(cid:17) − π x log x, (62)respectively. All the calculations are within the predicted ranges E i ( x ) = O ( x ). uler Phi Function and Largest Integer Function P n ≤ x ϕ ([ x/n ]). x P n ≤ x ϕ ([ x/n ]) 6 π − x log x Error E ( x )10 17 14 . 00 3 . . − . . 41 146 . . − . . − . P n ≤ x σ ([ x/n ]). x P n ≤ x σ ([ x/n ]) 6 − π x log x Error E ( x )10 39 37 . 88 1 . . 52 46 . . 80 714 . . 03 16112 . . 33 139776 . References [1] Apostol, Tom M. Introduction to analytic number theory . Undergraduate Texts inMathematics. Springer-Verlag, New York-Heidelberg, 1976.[2] Olivier Bordelles, Randell Heyman, Igor E. Shparlinski. On a sum involving theEuler function , arXiv:1808.00188.[3] F. Olver, M. McClain, et al, Editors. Digital Library of mathematical Functions. http://dlmf.nist.gov/5.4.el.[4] Zhai, Wenguang.