On the Unitary and Bi-Unitary Analogues of Generalized Ramanujan Sums

On the unitary and bi-unitary analogues of generalized Ramanujan sums

Isao Kiuchi and Sumaia Saad Eddin May 03, 2018

Abstract A positive integer d is called an unitary divisor of any positive integer n if d|n and (n, n/d) = 1, notation d||n. Let (k, n)∗ be the greatest divisor of the integer k which is an unitary divisor of n. Let (k, n)∗∗ be the greatest common unitary divisor of k and n. We introduce the following two functions X k X k t (j) = f(d)g , v (j) = f(d)g , k d k d d||(j,k)∗ dk(j,k)∗∗

for any arithmetical functions f and g. Here dk(j, k)∗ holds if and only if d|j and dkk. In this paper, we give some asymptotic formulas for the weighted averages of tk(j) and vk(j) with weights concerning the monomial factor, the Gamma function, and the Bernoulli polynomials. We also derive useful formulas for the unitary and the bi-unitary analogues of the gcd-sum function.

1 Introduction and statements of the results

Let (k, n) be the greatest common divisor of the integers k and n. In 1885, Ces`aropub- lished an important result on the arithmetic function, showing that for any arithmetic function f, we have n X X n P (n) := f ((k, n)) = f(d)φ . (1) d k=1 d|n In a special case when f = id, one can write n X X n (k, n) = dφ = (id ∗φ)(n). (2) d k=1 d|n

Mathematics Subject Classiﬁcation 2010: 11A25, 11N37, 11Y60. Keywords: Ramanujan’s sums; the gcd-sum function; unitary divisor; Euler totient function.

1 This latter sum is called by the gcd-sum function (sometime by Pallai’s arithmetic function) and due to Pallai (1937). Of course, the symbol ∗ denotes the Dirichlet con- P volution of two arithmetical functions f and g deﬁned by (f ∗ g)(n) = d|n f(d)g(n/d), for every positive integer n. Various generalizations and analogues of Eqs. (1) and (2) have been widely studied by many authors. For a nice survey see [5]. It is well known that the Ramanujan sum is deﬁned by

X k c (j) = dµ , k d d|(k,j) for any positive integers k and j. One of the most common generalization of the Ramanujan sum is due to Anderson-Apostol [1] and deﬁned by

X k s (j) := f(d)g , (3) k d d|(k,j) for any arithmetical functions f and g. In this paper, we introduce the unitary and bi-unitary analogues of the Anderson-Apostol sums. The next two subsections describe precisely our functions and results.

1.1 Unitary analogues A positive integer d is called an unitary divisor of any positive integer n if d|n and (n, n/d) = 1, notation d||n. Let (k, n)∗ be the greatest divisor of k which is an unitary divisor of n, namely (k, n)∗ = max {d ∈ N : d|k, d||n} . In 1989, T´oth[4] introduced the unitary analogue of Eq. (2) as follows

n ∗ X P (n) = (k, n)∗. (4) k=1 In that paper, he showed that the function P ∗(n) is multiplicative and that X n P ∗(n) = dφ∗ , (5) d d||n where φ∗ denotes the unitary analogue of Euler’s function and deﬁned by

∗ φ (n) = # {k ∈ N : 1 ≤ k ≤ n, (k, n)∗ = 1} . (6) In other words, φ∗(n) is rewritten in the form X n φ∗(n) = dµ∗ = (id ∗µ∗)(n). d d|n

2 Here µ∗(n) the unitary analogue of the M¨obiusfunction given by µ∗(n) = (−1)ω(n) where ω(n) is the he number of distinct prime factors of n.

We introduce the function tk(j) deﬁned by

X k t (j) = f(d)g , (7) k d d||(j,k)∗ such that dk(j, k)∗ holds if and only if d|j and dkk. The function tk(j) is an unitary analogue of Eq.(3). From the above, we immediately deduce the identity

k k/d X X k X t (j) = f(d)g 1 = (f ? g · id)(k). (8) k d j=1 dkk `=1

Here the symbol ? denotes the unitary convolution of two arithmetical functions f and P g deﬁned by (f ? g)(n) = d||n f(d)g(n/d), for every positive integer n. Eq.(8) is a generalization of Eq. (5). Moreover, we have

k X 1 X X f(d) t (j) = g(`). (9) k k d k≤x j=1 d`≤x (d,`)=1

Since φ is a multiplicative function, one can also see that

k X 1 X X f(d) g(`)` t (j) = . (10) φ(k) k φ(d) φ(`) k≤x j=1 d`≤x (d,`)=1

For any complex z we deﬁne the functions Bn(x) by the generating function

xz ∞ ze X Bn(x) = zn ez − 1 n! n=0 for |z| < 2π. The functions Bn(x) are known as Bernoulli polynomials, and the numbers Bn(0) are called Bernoulli numbers and are denoted by Bn.

Our ﬁrst goal of this paper is to derive some formulas of the weighted averages of tk(j) with weight function w concerning the monomial factor, the Gamma function Γ(.), and the Bernoulli polynomials Bn(.). This is

k X 1 X w(j)t (j) (11) W (k) k k≤x j=1

3 with certain weight function W . For any fixed positive integer r, we define the following function: k X 1 X T (x; f, g) := jrt (j), (12) r kr+1 k k≤x j=1 Then we prove that Theorem 1. For any positive real number x ≥ 2 and any fixed positive integer r, we have 1 X f(d) g(`) 1 X f(d) T (x; f, g) = + g(`) r 2 d ` r + 1 d d`≤x d`≤x (d,`)=1 (d,`)=1 [r/2] 1 X r + 1 X f(d) g(`) + B (13) r + 1 2m 2m d `2m m=1 d`≤x (d,`)=1 In a special case of Theorem 1 when f = f ∗ µ and g = 1, we get an useful formula for the unitary analogue of gcd-sum function, that is Corollary 1. Under the hypotheses of Theorem 1, we have 1 X (f ∗ µ)(d) 1 1 X (f ∗ µ)(d) T (x; f ∗ µ, 1) = + r 2 d ` r + 1 d d`≤x d`≤x (d,`)=1 (d,`)=1 [r/2] 1 X r + 1 X (f ∗ µ)(d) 1 + B . (14) r + 1 2m 2m d `2m m=1 d`≤x (d,`)=1 Now, we define the following two functions

k X 1 X j G (x) = log Γ t (j), f,g k k k k≤x j=1 and k−1 X 1 X j Y (x) = B t (j), f,g k m k k k≤x j=0 for any ﬁxed positive integer m. We also prove that Theorem 2. for any positive real number x ≥ 2 and any ﬁxed positive integer m, we have √ X f(d) X f(d) g(`) √ G (x) = log 2π g(`) − log 2π`, (15) f,g d d ` d`≤x d`≤x (d,`)=1 (d,`)=1 and X f(d) g(`) Y (x) = B (16) f,g m d `m d`≤x (d,`)=1

4 By taking f ∗µ in place of f and g = 1 in the above, we get the following interesting formulas.

Corollary 2. Under the hypotheses of Theorem 2, we have √ √ X (f ∗ µ)(d) X (f ∗ µ)(d) log 2π` G (x) = log 2π − , f∗µ,1 d d ` d`≤x d`≤x (d,`)=1 (d,`)=1 and X (f ∗ µ)(d) 1 Y (x) = B . f∗µ,1 m d `m d`≤x (d,`)=1

1.2 Bi-unitary analogues

Let (k, n)∗∗ be the greatest common unitary divisor of k and n, namely

(k, n)∗∗ = max {d ∈ N d||k, d||n} . In 2008, Haukkanen [3] deﬁned a generalization of the gcd-sum function as follows:

n ∗∗ X P (n) = (k, n)∗∗. (17) k=1 It is called by the bi-unitary gcd-sum function. Not surprisingly the study of P ∗∗(n) has a lot of similarities with that of P ∗(n) or even P (n). Haukkanen also showed that X n P ∗∗(n) = φ∗(d)φ , d , (18) d d||n where φ(x, d) is the Legendre function. More recently, T´oth [6] gave an asymptotic formula for the partial sum of P ∗∗(n). He proved that

X 1 P ∗∗(n) = Bx2 log x + O x2 , 2 n≤x where Y 3p − 1 Y (2p − 1)2 B = 1 − = ζ(2) 1 − . p2(p + 1) p4 p p Using the above and the partial summation, he deduced that

X P ∗∗(n) = Bx log x + O (x) . (19) n n≤x

5 By similar considerations, we introduce the bi-unitary analogue of Anderson-Apostol sums vk(j) deﬁned by X k v (j) = f(d)g . (20) k d dk(j,k)∗∗ With some careful calculations, one can check that

k k/(dδ) X X k X X v (j) = f(d)g µ(δ) 1 (21) k d j=1 dkk δ|d `=1

= (f · φ · id−1 ?g · id)(k). (22)

Then, we get k X 1 X X f(d) φ(d) v (j) = g(`), (23) k k d d k≤x j=1 d`≤x (d,`)=1

k X 1 X X f(d) g(`)` v (j) = . (24) φ(k) k d φ(`) k≤x j=1 d`≤x (d,`)=1

For any positive integer r, we consider the function Vr(x; f, g)

k X 1 X V (x; f, g) := jrv (j) (25) r kr+1 k k≤x j=1 to get the following result.

Theorem 3. For any positive real number x > 1 and any ﬁxed positive integer r, we have

f(1) X g(k) 1 X f(d) φ(d) V (x; f, g) = + g(`) r 2 k r + 1 d d k≤x d`≤x (d,`)=1 [r/2] 1 X r + 1 X f(d) φ1−2m(d) g(`) + B (26) r + 1 2m 2m d d1−2m `2m m=1 d`≤x (d,`)=1

As a consequence of Theorem 3, we deduce the following bi-unitary analogue of the gcd-sum function by replacing f by f ∗ µ and g = 1.

6 Corollary 3. Under the hypotheses of Theorem 3. For any arithmetical function f, we have

(f ∗ µ)(1) X 1 1 X (f ∗ µ)(d) φ(d) V (x; f ∗ µ, 1) = + r 2 k r + 1 d d k≤x d`≤x (d,`)=1 [r/2] 1 X r + 1 X (f ∗ µ)(d) φ1−2m(d) 1 + B (27) r + 1 2m 2m d d1−2m `2m m=1 d`≤x gcd(d,`)=1

Our second goal of this paper is to give asymptotic formulas of the weighted averages of vk(j) with weights concerning the Gamma function and the Bernoulli polynomials. Put k X 1 X j Gef,g(x) = log Γ vk(j), k k k≤x j=1 and k−1 X 1 X j Yef,g(x) = Bm vk(j). k k k≤x j=0 We prove that:

Theorem 4. Let the notation be as above, we have

√ X f(d)φ(d) Gef,g(x) = log 2π g(`) d2 d`≤x (d,`)=1 X g(k) √ 1 X f(d)Λ(d) g(`) − f(1) log 2πk − (28) k 2 d ` k≤x d`≤x (d,`)=1 where Λ is the von Mangoldt function. Furthermore, we have

X f(d) φ1−m(d) g(`) Yef,g(x) = Bm (29) d d1−m `m d`≤x (d,`)=1

As an application of Theorem 4, we take f ∗ µ in place of f and g = 1 into the above to get the interesting formulas

7 Corollary 4. Let the notation be as above, we have

k X 1 X j X Gef∗µ,1(x) := log Γ (f ∗ µ)(d) k k k≤x j=1 dk(j,k) ∗∗ √ √ X (f ∗ µ)(d)φ(d) X log 2πk = log 2π − f(1) d2 k d`≤x k≤x gcd(d,`)=1 1 X (f ∗ µ)(d)Λ(d) 1 − , (30) 2 d ` d`≤x gcd(d,`)=1 and k−1 X 1 X j X Yef∗µ,1(x) := Bm (f ∗ µ)(d) k k k≤x j=0 dk(j,k)∗∗

X (f ∗ µ)(d) φ1−m(d) 1 = B . (31) m d d1−m `m d`≤x gcd(d,`)=1

2 Proof of Theorems 1 and 3

2.1 Proof of Theorem 1

By the deﬁnition of tk(j), we have

k r k r X j X j X k t (j) = f(d)g k k k d j=1 j=1 d||k d|j k 1 X k X = f(d)g jr kr d d||k j=1 d|j k/d 1 X k X = drf(d)g `r kr d d||k `=1 We use the well known identity, see [2, Proposition 9.2.12],

N [r/2] X N r 1 X r + 1 `r = + B N r+1−2m (32) 2 r + 1 2m 2m `=1 m=0 for any positive integer N > 1, to obtain

k r [r/2] 1−2m X j 1 X k 1 X r + 1 X k k t (j) = f(d)g + B f(d)g k k 2 d r + 1 2m 2m d d j=1 d||k m=0 d||k

8 Using the fact that d||k is d|k and (d, k/d) = 1, we conclude that

k X 1 X 1 X f(d) g(`) 1 X f(d) jrt (j) = + g(`) kr+1 k 2 d ` r + 1 d k≤x j=1 d`≤x d`≤x (d,`)=1 (d,`)=1 [r/2] 1 X r + 1 X f(d) g(`) + B . r + 1 2m 2m d `2m m=1 d`≤x (d,`)=1

This completes the proof of Theorem 1.

2.2 Proof of Theorem 3

By the deﬁnition of vk(j), we have

k r k r X j X j X k v (j) = f(d)g k k k d j=1 j=1 d||k d||j k 1 X k X = f(d)g jr kr d d||k j=1 d||j k/d 1 X k X = drf(d)g `r kr d d||k `=1 (d,`)=1 k/d 1 X k X X = drf(d)g `r µ(q). kr d d||k `=1 q|d q|`

It follows that

k r k/d X j 1 X k X X v (j) = drf(d)g µ(q) `r k k kr d j=1 d||k q|d `=1 q|` k/(dq) 1 X k X X = drf(d)g µ(q)qr qr kr d 1 d||k q|d q1=1

9 Applying Eq. (32) to the inner sum on the right-hand side above, we ﬁnd that

k r X j 1 X k X v (j) = f(d)g µ(q) k k 2 d j=1 d||k q|d

[r/2] 1−2m 1 X r + 1 X k k X µ(q) + B f(d)g r + 1 2m 2m d d q1−2m m=0 d||k q|d P P Now, we notice that q|d µ(q) = 1 where d = 1. Otherwise q|d µ(q) = 0. Moreover, we have X µ(q) φ1−2m(d) = . q1−2m d1−2m q|d Thus, we get

k r [r/2] X j 1 1 X r + 1 v (j) = f(1)g(k)+ B (f · φ · id ? g · id )(k) k k 2 r + 1 2m 2m 1−2m 2m−1 1−2m j=1 m=0

Therefore, we obtain that

k X 1 X f(1) X g(k) 1 X f(d) φ(d) jrv (j) = + g(`) kr+1 k 2 k r + 1 d d k≤x j=1 k≤x d`≤x (d,`)=1 [r/2] 1 X r + 1 X f(d) φ1−2m(d) g(`) + B . r + 1 2m 2m d d1−2m `2m m=1 d`≤x (d,`)=1

This completes the proof.

3 Proof of Theorems 2 and 4

3.1 Proof of Theorem 2 Notice that k k/d X j X k X d` t (j) log Γ = f(d)g log Γ k k d k j=1 d||k `=1 Using the multiplication formula of Gauss–Legendre for the gamma function, see [2, Proposition 9.6.33] n n−1 Y j (2π) 2 Γ = √ , (33) n n j=1

10 we get

k X j log(2π) X k k t (j) log Γ = f(d)g k k 2 d d j=1 d||k log(2π) X k 1 X k k − f(d)g − f(d)g log . 2 d 2 d d d||k d||k

This leads to

k X 1 X j log(2π) X f(d) t (j) log Γ = g(`) k k k 2 d k≤x j=1 d`≤x (d,`)=1 log(2π) X f(d) g(`) 1 X f(d) g(`) log ` − − . 2 d ` 2 d ` d`≤x d`≤x (d,`)=1 (d,`)=1

This completes the proof of Eq. (15). Now, we have

k k−1 −1 X j X k Xd ` B t (j) = f(d)g B . m k k d m k/d j=0 dkk `=0

Using the following property of Bernoulli polynomial, see [2, Proposition 9.1.3 ]

n−1 X j Bm B = (34) m n nm−1 j=0 for any ﬁxed positive integer m, we get

k−1 X j Bm X k B t (j) = dm−1f(d)g . m k k km−1 d j=0 dkk

Therefore, we deduce that

k−1 X 1 X j X f(d) g(`) B t (j) = B . k m k k m d `m k≤x j=0 d`≤x (d,`)=1

The formula (16) is proved.

11 3.2 Proof of Theorem 4 Notice that

k k/d X j X k X d` v (j) log Γ = f(d)g log Γ k k d k j=1 d||k `=1 (`,d)=1 k/(dδ) X k X X qdδ = f(d)g µ(δ) log Γ d k d||k δ|d q=1

Using Eq. (33) and −Λ = µ · log ∗ 1, we get

k X j v (j) log Γ k k j=1 √ X k X k 1 X k X k = log 2π f(d)g µ(δ) − 1 − f(d)g µ(δ) log d dδ 2 d dδ d||k δ|d d||k δ|d √ X k k X µ(δ) √ X k X = log 2π f(d)g − log 2π f(d)g µ(δ) d d δ d d||k δ|d d||k δ|d 1 X k k X 1 X k X − f(d)g log µ(δ) + f(d)g µ(δ) log δ 2 d d 2 d d||k δ|d d||k δ|d √ X k k φ(d) √ 1 X k = log 2π f(d)g − log 2πkf(1)g(k) − f(d)g Λ(d). d d d 2 d d||k d||k

Thus, we ﬁnd that

k X 1 X j √ X f(d)φ(d) v (j) log Γ = log 2π g(`) k k k d2 k≤x j=1 d`≤x (d,`)=1 X g(k) √ 1 X f(d)Λ(d) g(`) − f(1) log 2πk − . k 2 d ` k≤x d`≤x (d,`)=1

12 This completes the proof of Eq. (28). For Eq. (29), we note that

k k−1 −1 X j X k Xd d` B v (j) = f(d)g B m k k d m k j=0 d||k `=0 (`,d)=1 k −1 X k X dδX dδj = f(d)g µ(δ) B d m k d||k δ|d j=0 1−m X k k φ1−m(d) = B f(d)g . m d d d1−m dkk

We therefore conclude that k−1 X 1 X j X f(d) φ1−m(d) g(`) B v (j) = B , k m k k m d d1−m ` k≤x j=0 d`≤x (d,`)=1 which gives the desired result.

Acknowledgement

The second author is supported by the Austrian Science Fund (FWF) : Project F5507- N26, which is part of the special Research Program “ Quasi Monte Carlo Methods : Theory and Application”.

References

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[2] H. Cohen, Number Theory, vol II: Analytic and modern tools, Graduate Texts in Mathematics, 240, Springer, (2007).

[3] P. Haukkanen, On a gcd-sum function, Aequationes. Math 76 (2008), 168–178.

[4] L. T´oth, The unitary analogue of Pillai’s arithmetical function II, Notes Number Theory Discrete Math. 2 (1996), 40–46.

[5] L. T´oth,A survey of gcd-sum functions, J. Integer Sequences 13 (2010), Article 10.8.1.

[6] L. T´oth,On the bi-unitary analogues of Euler’s arithmetical function and the gcd- sum function, J. Integer Sequences 12 (2009), Article 09.5.2.

13 Isao Kiuchi: Department of Mathematical Sciences, Faculty of Science, Yamaguchi University, Yoshida 1677-1, Yamaguchi 753-8512, Japan. e-mail: [email protected]

Sumaia Saad Eddin: Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University, Altenbergerstrasse 69, 4040 Linz, Austria. e-mail: sumaia.saad [email protected]