COUNTING INVERTIBLE SUMS OF SQUARES MODULO n AND A

arXiv:1403.7878v1 [math.NT] 31 Mar 2014

COUNTING INVERTIBLE SUMS OF SQUARES MODULO n AND A NEW GENERALIZATION OF EULER TOTIENT FUNCTION ´ ´ MAR´IA GRAU, AND ANTONIO M. OLLER-MARCEN ´ CATALINA CALDERON, JOSE

Abstract. In this paper we introduce and study a family Φk of arithmetic functions generalizing Euler’s totient function. These functions are given by the number of solutions to the equation gcd(x21 +···+x2k , n) = 1 with xi ∈ Z/nZ which, for k = 2, 4 and 8 coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over Z/nZ. We prove that Φk is multiplicative for every k, we obtain an explicit formula for Φk (n) in terms of the prime-power decomposition of n and we study some properties that extend well-known results for ϕ. As a tool we study the multiplicative arithmetic function that counts the number of solutions to x21 + · · · + x2k ≡ λ (mod n) for λ coprime to n, thus extending an old result that dealt only with the prime n case.

1. Introduction Euler’s totient function ϕ is one of the most famous arithmetic functions used in number theory. Recall that ϕ(n) is defined as the number of positive integers less than or equal to n that are coprime to n. Many generalizations Euler’s function are known. See, for instance [4, 6, 7, 11, 14] or the special chapter on this topic in [13]. Among these generalizations, the most significant isQprobably the so-called Jordan’s totient function [1, 2, 16] defined as Jk (n) := nk p|n (1 − p−k ). In this paper we introduce and study a new generalization of ϕ. In particular, given k ∈ N we define Φk (n) := card {(x1 , · · · , xk ) ∈ Zk : 0 ≤ xi < n and gcd(x21 + · · · + x2k , n) = 1}. Clearly Φ1 (n) = ϕ(n) and it is the order of the group of units of the ring Z/nZ. On the other hand, Φ2 (n) is the so-called GIphi which computes the number of Gaussian integers in a reduced system modulo n. In the same way, Φ4 (n) and Φ8 (n) compute, respectively, the number of invertible quaternions and octonions over Z/nZ. In order to study the function Φk we need to focus on the functions ρk,λ (n) := card {(xi , . . . , xk ) ∈ (Z/nZ)k : x21 + · · · + x2k ≡ λ (mod n)} which count the number of points on hyperspheres in (Z/nZ)k and, in particular, in the case gcd(λ, n) = 1. These functions were already studied in the case when n is an odd prime by V.H. Lebesgue in 1837. In particular he proved the following result [3, Chapter X]. 1

2

´ ´ MAR´IA GRAU, AND ANTONIO M. OLLER-MARCEN ´ CATALINA CALDERON, JOSE

Proposition 1. Let p be an odd prime and let k, λ be positive integers with p ∤ λ. Put t = (−1)(p−1)(k−1)/4 p(k−1)/2 and l = (−1)k(p−1)/4 p(k−2)/2 . Then,  k−1  + t, If k is odd and λ is a quadratic residue modulo p; p k−1 ρk,λ (p) = p − t, If k is odd and λ is a not quadratic residue modulo p;   k−1 p − l, If k is even.

The paper is organized as follows. First of all, in Section 2 we study the values of ρk,λ (n) in the case gcd(λ, n) = 1, thus generalizing Lebesgue’s work. In Section 3 we study the functions Φk , in particular we prove that they are multiplicative and we give a closed formula for Φk (n) in terms of the prime-power decomposition of n. Finally, we close our work presenting some properties of the functions Φk that generalize known properties of Euler’s totient function (which are recovered if k = 1) and suggesting some ideas that leave the door open for future work. 2. Counting points on hyperspheres (mod n) Due to the Chinese Remainder Theorem, the function ρk,λ is multiplicative; i.e., if n = pr11 · · · prmm , then ρk,λ (n) = ρk,λ (pr11 ) · · · ρk,λ (prmm ). Hence, we can restrict ourselves to the case when n = ps is a prime-power. Moreover, since in this paper we focus on the case gcd(λ, n) = 1, we will always assume that p ∤ λ. The following result will allow us to extend Lebesgue’s work to the odd prime-power case. Lemma 1. Let p be an odd prime and let s ≥ 1. If p ∤ λ, then ρk,λ (ps ) = p(s−1)(k−1) ρk,λ (p).

Proof. It is easily seen that any solution to the congruence x2i +· · · x2k ≡ λ (mod ps+1 ) must be of the form (a1 + t1 ps , . . . , ak + tk ps ) for some (a1 , . . . , ak ) such that a21 +· · · a2k ≡ λ (mod ps ). Now, (a1 +t1 ps )2 +· · ·+(ak +tk ps )2 ≡ λ (mod ps+1 ) if and only if 2a1 t1 +· · ·+2ak tk ≡ −K (mod p), where K is such that a21 +· · · a2k = Kps +λ. Since ai 6≡ 0 (mod p) for some i ∈ {1, . . . , k}, it follows that there are exactly pk−1 possibilities for (t1 , . . . , tk ) and the result follows inductively.  If p = 2 we have a similar result. Lemma 2. Let s ≥ 3 and let λ ≥ 1 be odd. Then, ρk,λ (2s ) = 2(s−3)(k−1) ρk,λ (8). Proof. If s ≥ 3 it can be easily seen that any solution to the congruence x21 + · · · + x2k ≡ λ (mod 2s+1 ) must be of the form (a1 + t1 2s−1 , . . . , ak + tk 2s−1 ) for some (a1 , . . . , ak ) such that a21 + · · · a2k ≡ λ (mod 2s ). Now, (a1 + t1 2s−1 )2 + · · · + (ak + tk 2s−1 )2 ≡ λ (mod 2s−1 ) if and only if a1 t1 + · · · ak tk ≡ −K (mod 2), where K is such that a21 + · · · a2k = K2s + λ. Since ai must be odd fr some i ∈ {1, . . . , k}, it follows that there are exactly 2k−1 possibilities for (t1 , . . . , tk ) and the result follows inductively.  As we have just seen, unlike when p is an odd prime, the recurrence is now based on ρk,λ (23 ). Hence, the cases s = 1, 2, 3; i.e., n = 2, 4, 8, must be studied separately. In order to do so, the following general result will be useful.

COUNTING INVERTIBLE SUMS OF SQUARES MODULO n AND A NEW GENERALIZATION OF EULER TOTIENT FUNCTION 3

Proposition 2. Let k, λ ≥ 1 and let n be a positive integer. Then, ρk,λ (n) =

n−1 X

ρ1,i (n)ρk−1,λ−i (n).

i=0

Proof. Let (x1 , . . . , xk ) ∈ A(k, λ, n); i.e., x21 + · · · + x2k ≡ λ (mod n). Then, for some i ∈ {0, . . . , n − 1} we have that x21 ≡ i (mod n) and x22 + · · · + x2k ≡ λ − i (mod n) and hence the result.    Now, given k, n ≥ 1 let us define the matrix M (n) = ρ1,i−j (n) . If 0≤i,j≤n−1   we consider the column vector Rk (n) = ρk,i (n) , then Lemma 2 leads to 0≤i≤n−1

the following recurrence relation:

Rk (n) = M (n) · Rk−1 (n).

In the following proposition we use this recurrence relation to compute ρk,λ (2s ) for s = 1, 2, 3 and odd λ. Proposition 3. Let k be a positive integer. Then: i) ρk,1 (2) = 2k−1 ,
3k ii) ρk,1 (4) = 4k−1 + 2 2 −1 sin πk 4
, 3k iii) ρk,3 (4) = 4k−1 − 2 2 −1 sin πk 4 ,


k 1 1 k 2k−3 +1 2 + 2 2 sin πk iv) ρk,1 (8) = 2 , π(k + 1) − 2 cos (3πk + π) + 2 sin 4 4 4  



k , − 2 cos 41 π(k + 1) + cos 43 π(k + 1) v) ρk,3 (8) = 22k−3 2k − 2 2 +1 sin πk 4 


 k πk 1 1 +1 k 2k−3 2 + 2 2 sin 4 − 2 sin 4 π(k + 1) + 2 cos 4 (3πk + π) , vi) ρk,5 (8) = 2 


k 1 1 (3πk + π) + 2 cos π(k + 1) − 2 sin . vii) ρk,7 (8) = 22k−3 2k − 2 2 +1 sin πk 4 4 4 Proof. First of all, observe that

 1 M (2) = 1

 2  2 1 , M (4) =  0 1 0

0 2 2 0

0 0 2 2

 2 0 , 0 2

 2 4  0  0 M (8) =  2  0  0 0

0 2 4 0 0 2 0 0

0 0 2 4 0 0 2 0

0 0 0 2 4 0 0 2

2 0 0 0 2 4 0 0

0 2 0 0 0 2 4 0

0 0 2 0 0 0 2 4

 4 0  0  2 . 0  0  0 2

Let us compute ii). We know that Rk (4) = M (4) · Rk−1 (4). Hence, since the eigenvalues of M (4) are {4, 2 + 2i, 2 − 2i, 0}, we know that ρk,1 (4) = C1 4k + C2 (2 + 2i)k + C3 (2 − 2i)k .

In order to compute C1 , C2 and C3 it is enough to observe that ρ1,1 (4) = 2, ρ2,1 (4) = 8 and ρ3,1 (4) = 24. Hence: 4C1 + (2 + 2i)C2 + (2 − 2i)C3 = 2, 16C1 + 8iC2 − 8iC3 = 8,

64C1 − (16 − 16i)C2 − (16 + 16i)C3 = 24.

4

´ ´ MAR´IA GRAU, AND ANTONIO M. OLLER-MARCEN ´ CATALINA CALDERON, JOSE

And we get
3k 1 k 4 − i(2 + 2i)k + i(2 − 2i)k = 22k−2 + 2 2 −1 sin ρk,1 (4) = 4



πk 4



,

as claimed. To compute the other cases note that the eigenvalues of M (2) are {0, 2} while the eigenvalues of M (8) are n o √ √ √ √ 8, 4 + 4i, 4 − 4i, 2(−2 − 2i), 2(2 + 2i), 2(−2 + 2i), 2(2 − 2i), 0 . Thus, in each case we only need to compute the corresponding initial conditions and constants. The final results have been obtained with the help of Mathematica “ComplexExpand” command.  3. Counting invertible sums of squares modulo n Given positive integers k, n, this section is devoted to computing the number of solutions of gcd(x21 + · · · + x2k , n) = 1, that we shall denote by Φk (n). First of all, let us define the set [ A(k, λ, n). Ak (n) := 1≤λ≤n gcd(λ,n)=1

Hence, Φk (n) = card Ak (n) and, since the union is clearly disjoint, it follows that X ρk,λ (n). Φk (n) = 1≤λ≤n gcd(λ,n)=1

The following result shows the multiplicativity of Φk for every positive k. Theorem 1. Let k be a positive integer. Then Φk is multiplicative; i.e., Φk (mn) = Φk (m)Φk (n) for every m, n ∈ Z such that gcd(m, n) = 1. Proof. Let us define a map F : Ak (m) × Ak (n) −→ Ak (mn) by F ((a1 , . . . , ak ), (b1 , . . . , bk )) = (na1 + mb1 , . . . , nak + mbk ). Note that if (a1 , . . . , ak ) ∈ Ak (m), then a21 + · · · + a2k ≡ λ1 (mod m) for some λ1 with gcd(λ1 , m) = 1. In the same way, if (b1 , . . . , bk ) ∈ Ak (n), then b21 +· · ·+b2k ≡ λ2 (mod n) for some λ2 with gcd(λ2 , n) = 1. Consequently, (na1 + mb1 )2 + · · · + (nak + mbk )2 = n2 (a21 + · · · + a2k ) + m2 (b21 + · · · + b2k ) + 2mn(b1 a1 + · · · + bk ak ) ≡ ≡ n2 λ1 + m2 λ2

(mod mn).

Since it is clear that gcd(n2 λ1 +m2 λ2 , mn) = 1, it follows that (na1 +mb1 , . . . , nak + mbk ) ∈ Ak (mn) and thus F is well-defined. Now, let (c1 , . . . , ck ) ∈ Ak (mn). Then c21 + · · · + c2k ≡ λ (mod mn) for some λ such that gcd(λ, mn) = 1. Let us define ai ≡ ci (mod m) and bi ≡ ci (mod n) for every i = 1, . . . , k. It follows that (a1 , . . . , ak ) ∈ Ak (m), b1 , . . . , bk ∈ Ak (n) and, moreover, F ((a1 , . . . , ak ), (b1 , . . . , bk )) = (c1 , . . . , ck ). Hence, F is surjective. Finally, assume that (na1 + mb1 , · · ·, nak + mbk ) ≡ (nα1 + mβ1 , · · ·, nαk + mβk ) (mod mn)

COUNTING INVERTIBLE SUMS OF SQUARES MODULO n AND A NEW GENERALIZATION OF EULER TOTIENT FUNCTION 5

for some (a1 , . . . , ak ), (α1 , . . . , αk ) ∈ Ak (m) and for some (b1 , . . . , bk ), (β1 , . . . , βk ) ∈ Ak (n). Then, for every i = 1, . . . , k we have that nai +mbi ≡ nαi +mβi (mod mn). From this, it follows that ai ≡ αi (mod m) and that bi ≡ βi (mod n) for every i and hence F is injective. Thus, we have proved that F is bijective and the result follows.  Since we know that Φk is multiplicative, we just need to compute its values over prime-powers. We do so in the following result. Proposition 4. Let k, r be positive integers. i) Φk (2r ) = ϕ(2kr ). ii) If p is an odd prime,  kr  if k ϕ(p ), r kr kr−k/2 Φk (p ) = ϕ(p ) − ϕ(p ), if k   ϕ(pkr ) + ϕ(pkr−k/2 ), if k

Then:

is odd; is even and 4 | k or 4 | p − 1; is even, 4 ∤ k and 4 ∤ p − 1.

Proof. i) If r = 1, 2, 3 the result readily follows from Proposition 3 by simple computation. Now, if r > 3 we can apply Lemma 2 to obtain that X X Φk (2r ) = ρk,i (2r ) = 2(r−3)(k−1) ρk,i (8) 1≤i≤2r 2∤i

=2

1≤i≤2r 2∤i

(r−3)(k−1)

2r−3 X−1 j=0

= 2(r−3)(k−1) 2r−3

X

ρk,i (8) =

8j+1≤i≤8(j+1)−1 2∤i

X

ρk,i (8) =

1≤i≤7 2∤i

= 2(r−3)(k−1) 2r−3 23k−1 = 2rk−1 = ϕ(2kr ). ii) Due to Lemma 1 it can be seen, as is the previous case, that r

Φk (p ) = p

k(r−1)

p−1 X

ρk,i (p).

i=1

Thus, it is enough to apply Proposition 1.  Finally, we summarize the previous work in the following result. Corollary 1. Let k, n be positive integers. Then,  k−1 n ϕ(n),     Y   1  k−1  1 − k/2 , ϕ(n)  n p 26=p|n Φk (n) =   Y Y  1  nk−1 ϕ(n) 1 +   k/2  p   p|n p|n p≡3

(mod 4)

p≡1

(mod 4)

if k is odd; if k ≡ 2 (mod 4);   1 1 − k/2 , p

if 4 | k.

6

´ ´ MAR´IA GRAU, AND ANTONIO M. OLLER-MARCEN ´ CATALINA CALDERON, JOSE

Proof. Just apply the multiplicativity of Φk and observe that k

ϕ(pkr ) − ϕ(pkr−k/2 ) = pkr− 2 −1 (p − 1)(pk/2 − 1), k

ϕ(pkr ) + ϕ(pkr−k/2 ) = pkr− 2 −1 (p − 1)(pk/2 + 1).



When k is a multiple of 4, Φk is closely related to Jk/2 . The following result makes this relation explicit. Its proof, that we omit, follows from the previous corollary recalling the definition of Jordan’s totient function Jk . Proposition 5. Let k be a multiple of 4. Then, k

k

Φk (n) = n 2 −1 Jk/2 (n)ϕ(n)

2 2 − 1 − mod(n, 2) k

22

.

Moreover, if k/4 is odd, we have that Φk (n) 2k/4 − 1 − mod(n, 2) = nJk/2 (n) Φk/4 (n) 2k/4 If k = 4 recall that Φ4 (n) is the number of units in the ring H(Z/nZ). If, in addition, n is odd then Φ4 (n) = nJ2 (n)ϕ(n) which is the well-known formula for the number of regular matrices in the ring M2 (Z/nZ). Of course, this is not a surprise since it is known that for an odd n the rings H(Z/nZ) and M2 (Z/nZ) are isomorphic [5]. 4. Some properties of Φk In this section we will present some properties of the function Φk which are counterparts of known properties of Euler’s totient function (recall that Φ1 = ϕ). 4.1. The first elementary properties of Φk . Proposition 6. Let m, n be positive integers such that n | m. Then, Φk (n) | Φk (m) for every k ≥ 1.

Proof. Note that, if p is prime and r ≤ s, Proposition 4 implies that Φk (pr ) | Φk (ps ). Hence, the result follows because Φk is multiplicative.  The function Φk is multiplicative, but not completely multiplicative. The following result makes this clear. Proposition 7. Let m, n, k be positive integers and let d = gcd(m, n). Then, Φk (mn) = Φk (m)Φk (m)

dk . Φk (d)

Proof. Let p be a prime and let r1 , r2 be positive integers with r1 < r2 . If we put r = r1 + r2 , Proposition 4 leads to Φk (pr ) = Φk (pr1 )Φk (pr2 )

pkr1 . Φk (pr1 )

Thus, it is enough to consider the prime-power decomposition of m and n to complete the proof.  Finally, if we apply Proposition 7 in the case m = n we get the following.

COUNTING INVERTIBLE SUMS OF SQUARES MODULO n AND A NEW GENERALIZATION OF EULER TOTIENT FUNCTION 7

Corollary 2. Let n, k be positive integers. Then, Φk (nm ) = nkm−k Φk (n). 4.2. Asymptotic growth of Φk . We now focus on the asymptotic behavior of Φk (n)/nk ; i.e., on the average number of points in [1, n]k such that its sum of squares is coprime to n. We see that this behavior is independent of k ans, moreover, that it is the same as that of ϕ(n)/n. See [8] for results on the asymptotic growth of Euler ϕ function and its limits. Proposition 8. For every positive integer k we have that: lim sup

Φk (n) = 1, nk

lim inf

Φk (n) = 0. nk

Proof. It is enough to apply Corollary 1 together with the known results for ϕ(n)/n.  4.3. Divisor sum and M¨ obius inversion formula. The following identity is well-known: X ϕ(n) = n. S(n) := d|n

It follows from the fact that S is multiplicative and S(pr ) = pr for every primepower pr . In terms of Dirichlet’s convolution this identity can be written as S = U ∗ ϕ = N , where the functions U and N are given by U (n) = 1 and N (n) = n for every n ∈ N. Equivalently, using M¨ obius inversion formula, the previous identity can be written as ϕ = µ ∗ N with µ the M¨ obius function. If we now define X Φk (n), Sk (n) := (U ∗ Φk )(n) = d|n

we obtain again a multiplicative function whose values over prime-powers are given in the following result. Proposition 9. Let k be a positive integer. Then, 2kr − 1 i) Sk (2r ) = 1 + ϕ(2k ) k . 2 −1 ii) If p is an odd prime:  kr −1  k p   1 + ϕ(p ) , if k is odd;  k  p −1     pkr − 1 Sk (pr ) = 1 + ϕ(pk ) − ϕ(pk/2 ) k , if k is even and 4 | k or 4 | p − 1;  p −1     kr  p −1   1 + ϕ(pk ) + ϕ(pk/2 ) k , if k is even, 4 ∤ k and 4 ∤ p − 1. p −1

Proof. It is a direct consequence of Proposition 4.



Now, if σ is de usual divisor function, it is also known that S = U ∗ ϕ = N = µ ∗ σ. This P result can be generalized if k is odd in the following way (recall that σk (n) = d|n dk ).

8

´ ´ MAR´IA GRAU, AND ANTONIO M. OLLER-MARCEN ´ CATALINA CALDERON, JOSE

Proposition 10. Let k be an odd integer. Then, X µ(d)dk−1 σk (n/d.) = (nk−1 µ ∗ σk )(n) Sk (n) = d|n

Proof. First of all, observe that Corollary 1 implies that X X X dk−1 ϕ(d)U (n/d). dk−1 ϕ(d) = Φk (n) = Sk (n) = d|n

d|n

d|n

Hence, we have that Sk = N k−1 ϕ ∗ U = N k−1 (µ ∗ N ) ∗ U = (N k−1 µ ∗ N k ) ∗ U = N k−1 µ ∗ (N k ∗ U ) = = N k−1 µ ∗ σk ,

as claimed.



4.4. The average order of Φk . The average order of ϕ(n) is well-known [17]. Namely, X 3 ϕ(n) ≈ 2 x2 . π n≤x

In fact, the best known asymptotic formula is currently: X 3 ϕ(n) = 2 x2 + O(x(log(x)2/3 (log log x)4/3 ). π n≤x

If k is odd we can give an easy generalization of this fact. Theorem 2. Let k ≥ 1 be an odd integer. Then X 6 xk+1 + O(xk log2/3 x(log log x)4/3 ) Φk (n) = 2 π (k + 1) n≤x

Proof. For k = 1 we have X

ϕ(n) =

n≤x

3 2 x + O(x log2/3 x(log log x)4/3 ). π2

Let k > 1 be an odd integer. As Φk (n) = nk−1 ϕ(n) it is enough to apply the Abel′ s summation formula. Thus we obtain X

n≤x

 3 2 2/3 4/3 x + O(x log x(log log x) ) xk−1 − Φk (n) = π2 

−(k − 1) =

Z

1

x



 3 2 2/3 4/3 t + O(x log t(log log t) ) tk−2 dt π2

6 xk+1 + O(xk log2/3 x(log log x)4/3 ). π 2 (k + 1) 

COUNTING INVERTIBLE SUMS OF SQUARES MODULO n AND A NEW GENERALIZATION OF EULER TOTIENT FUNCTION 9

5. Conclusions and further work The generalization of ϕ that we have presented in this paper is possibly one of the closest to the original idea which consists of counting units in a ring. In addition, both the elementary and asymptotic properties of Φk extend those of ϕ in a very natural way. There are many other results regarding ϕ that have not been considered here but that, nevertheless, may have their extension to Φk . We now present some ideas in that direction. 5.1. Dirichlet series for Φk . The Dirichlet series for ϕ(n) may be written in terms of the Riemann zeta function as: ∞ X ϕ(n) ζ(s − 1) = . s n ζ(s) n=1 Again, if k is odd, we easily get the following result. Proposition 11. Let k be an odd integer. Then, ∞ X Φk (n) ζ(s − k) = . s n ζ(s + 1 − k) n=1 Proof. If k is odd recall that Φk (n) = nk−1 ϕ(n).



5.2. Menon’s identity. In 1965 P. Kesava Menon [12] proved the following identity: X gcd(k − 1, n) = d(n)ϕ(n), k|n;gcd(k,n)=1

where d(n) denotes the number of divisors of n. This identity has been generalized in several ways [9, 10, 15]. Our work suggests the following generalization: X gcd(x21 + ... + x2k − 1, n) = Γk (n)Φk (n), gcd(x21 +...+x2k ,n)=1

where Γk (n) is a multiplicative function to be found. References [1] Dorin Andrica and Mihai Piticari. On some extensions of Jordan’s arithmetic functions. Acta Univ. Apulensis Math. Inform., (7):13–22, 2004. [2] Leonard Eugene Dickson. History of the theory of numbers. Vol. I: Divisibility and primality. Chelsea Publishing Co., New York, 1966. [3] Leonard Eugene Dickson. History of the theory of numbers. Vol. II: Diophantine analysis . Chelsea Publishing Co., New York, 1966. [4] P. G. Garcia and Steve Ligh. A generalization of Euler’s ϕ-function. Fibonacci Quart., 21(1):26–28, 1983. [5] Miguel-C. Grau, J.M. and A.M. Oller-Marc´ en. On the structure of quaternion rings over Z/nZ. arXiv:1402.0956 [math.RA], (1). [6] Oller-Marc´ en A.M. Rodr´ıguez M. Grau, J.M. and D. Sadornil. Fermat test with gaussian base and gaussian pseudoprimes. arXiv:1401.4708 [math.NT], (1). [7] P. Hall. The eulerian functions of a group. Quarterly Journal of Mathematics, os-7(1):134– 151, 1936. Cited By (since 1996):98. [8] G. H. Hardy and E. M. Wright. An introduction to the theory of numbers. Oxford University Press, Oxford, sixth edition, 2008. Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles. [9] P. Haukkanen and J. Wang. A generalization of Menon’s identity with respect to a set of polynomials. Portugal. Math., 53(3):331–337, 1996.

10

´ ´ MAR´IA GRAU, AND ANTONIO M. OLLER-MARCEN ´ CATALINA CALDERON, JOSE

[10] Pentti Haukkanen. Menon’s identity with respect to a generalized divisibility relation. Aequationes Math., 70(3):240–246, 2005. [11] Jerzy Kaczorowski. On a generalization of the Euler totient function. Monatsh. Math., 170(1):27–48, 2013. P [12] P. Kesava Menon. On the sum (a − 1, n), [(a, n) = 1]. J. Indian Math. Soc. (N.S.), 29:155–163, 1965. [13] J. S´ andor and B. Crstici. Handbook of number theory. II. Kluwer Academic Publishers, Dordrecht, 2004. [14] R. Sivaramakrishnan. The many facets of Euler’s totient. II. Generalizations and analogues. Nieuw Arch. Wisk. (4), 8(2):169–187, 1990. [15] Marius T˘ arn˘ auceanu. A generalization of Menon’s identity. J. Number Theory, 132(11):2568– 2573, 2012. [16] S. Thajoddin and S. Vangipuram. A note on Jordan’s totient function. Indian J. Pure Appl. Math., 19(12):1156–1161, 1988. [17] Arnold Walfisz. Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte, XV. VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. ´ ticas, Universidad del Pa´ıs Vasco, Facultad de Ciencia y Departamento de Matema Tecnolog´ıa, Barrio Sarriena, s/n, 44980 Leioa, Spain E-mail address: [email protected] ´ ticas, Universidad de Oviedo, Avda. Calvo Sotelo, s/n, Departamento de Matema 33007 Oviedo, Spain E-mail address: [email protected] Centro Universitario de la Defensa, Ctra. de Huesca, s/n, 50090 Zaragoza, Spain E-mail address: [email protected]