- Email: [email protected]
Cesàro's formula in number fields
Accepted Manuscript Cesàro’s formula in number fields
C. Miguel
PII: DOI: Reference:
S0022-314X(17)30209-3 http://dx.doi.org/10.1016/j.jnt.2017.05.007 YJNTH 5772
To appear in:
Journal of Number Theory
Received date: Revised date: Accepted date:
1 February 2017 5 May 2017 6 May 2017
Please cite this article in press as: C. Miguel, Cesàro’s formula in number fields, J. Number Theory (2017), http://dx.doi.org/10.1016/j.jnt.2017.05.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
00 (2017) 1–6
Ces`aro’s formula in number fields C. Miguel Instituto de Telecomunica¸co˜ es, Beira Interior University, Department of Mathematics, Covilh˜a, Portugal, Email: [email protected]
Abstract We give an extension of a theorem of Ces`aro from the rational integers to the ring of integers of an arbitrary number field. This extension is used to generalize Pillai’s function to number fields. Keywords: Dedekind zeta function, Euler totient function, Gcd-sum function, Number field, Pillai function 2010 MSC: 11R42, 11R99
1. Introduction A theorem of Ces`aro (see e.g. [5], [8, p.127], [17]) states that for every natural number n ∈ N = {1, 2, . . .} and any arithmetical function f we have n f ((i, n)) = f (d)ϕ(n/d), (1) i=1
d|n
where (i, n) denotes the greatest common divisor of i and n, and ϕ is the Euler totient function. There is in the literature a large number of generalizations and analogues of Ces`aro theorem. For a rich and extensive survey concerning generalizations of Ces`aro theorem the reader is referred to [10]. Since rings of integers in a number field are natural generalizations of the rational integers, the question natural arises as to whether an analogous statement could be made for the ring of integers in a number field. Recently, there has been considerable interest in extending arithmetical identities from the rational integers to a more general setting (see e.g. [11], [13]). The aim of this paper is to extend Ces`aro theorem to the ring OK of integers in a number field K. Instead of working with elements in OK , where unique factorization can fail, we will work with ideals. As is well known using ideals in place of elements we can save unique factorization. More precisely, every nonzero ideal n of OK can be uniquely written in the form n = pα1 1 . . . pαs s , where p1 , . . . , p s are distinct nonzero prime ideals and α1 , . . . , α s are positive rational integers (see e.g. [14, p.8]). Unique factorization of ideals in OK permits calculations that are analogous to some familiar manipulations involving ordinary integers. In particular we can define the concept of arithmetical function on the set of ideals of OK . A real or complex-valued function defined on the set of ideals of the ring of integers in a number field is called an arithmetical function. As a very simple example, consider a nonzero ideal n of OK , then the generalized Euler totient function, which is denoted by ϕK (n), is defined to be the order of the multiplicative group of units in the factor ring OK /n, denoted by U(OK /n), with the convention that ϕK (OK ) = 1. That is, 1 i f n = OK , ϕK (n) = |U(OK /n)| otherwise. 1
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Note that, for each nonzero ideal n of OK , the residue class ring OK /n is finite (see e.g. [15, p.120]). Thus, ϕK is a finite valued function. Our main result is the following. Theorem 1.1. Let OK be the ring of integers in a number field K, and let n be a nonzero ideal of OK . Let ψ : OK → OK /n be the canonical epimorphism. Assume that n factorizes as n = pα1 1 . . . pαs s , where p1 , . . . , p s are distinct nonzero prime ideals and α1 , . . . , α s are positive rational integers. Then, for any arithmetical function f we have f (ψ−1 (i) + n) = f (pβ11 . . . pβs s )ϕK (pα1 1 −β1 . . . pαs s −βs ). (2) 0≤βi ≤αi
i∈OK /n
Note that, in view of the fact that OK /n is finite, the sum on the left is well-defined. By the unique factorization of ideals in OK it follows that the right side of equation (2) is equal to ab=n f (a)ϕK (b). 2. Some lemmas In this section we assemble the tools that we need to prove our results. Given a nonzero ideal n of OK , the group of units U(OK /n) acts naturally on the additive group of OK /n by left multiplication. Our next task is to describe and find the size of the orbits of this action. Let n = pα1 1 . . . pαs s be the prime factorization of the ideal n. By the Chinese remainder theorem for rings (see [9, p.265]) we obtain an isomorphism OK /n OK /pα1 1 ⊕ · · · ⊕ OK /pαs s .
(3)
U(OK /pαi i ),
Hence, the unit group U(OK /n) is the direct product of the groups for i = 1, . . . , s. In other words, the generalized Euler totient function ϕK is multiplicative. Consequently, we can limit ourselves to the case where the ideal n is a power of a nonzero prime ideal. So, let us assume that n = pα , where p is a nonzero prime ideal in OK and α is a positive rational integer. Note that, the ring OK /pα is a local ring with maximal ideal p/pα and all of its ideals are of the form pβ /pα , where 0 ≤ β ≤ α. Since every ideal in OK /pα is principal [2, p.99] it follows that the lattice of ideals of OK /pα is the chain 0 = aα ⊂ aα−1 ⊂ aα−2 ⊂ . . . ⊂ a ⊂ a0 = OK /pα , where a ∈ p/pα is a fixed generator of the maximal ideal p/pα . For a nonzero element x ∈ OK /pα let w(x) denote the largest integer m for which x ∈ am . Note that, if w(x) = m, then x may be uniquely represented in the form x = uam , where u is a unit in OK /pα . It is clear that the orbit of x is given by o(x) = U(OK /pα )aw(x) . Hence, the orbits of this action give rise to the following partition of OK /pα OK /pα = {0} ∪ U(OK /pα ) ∪ U(OK /pα )a ∪ U(OK /pα )a2 ∪ . . . ∪ U(OK /pα )aα−1 .
(4)
Our next goal is to find the size of each orbit. Note that p/pα is an additive subgroup of OK /pα of index N(p) = |OK /p|, the norm of the ideal p. Let C1 , . . . , C N(p) be the distinct cosets of p/pα in OK /pα . A set R = {u1 , . . . , uN(p) }, where ui ∈ Ci , for i = 1, . . . , N(p), is called a system of representatives for the field OK /p. Note that, if ψ : OK /pα → OK /p is the canonical epimorphism, then for each z ∈ OK /p there is a unique y ∈ R such that ψ(y) = z. Lemma 2.1. Let OK be the ring of integers in a number field K. Let p be a prime ideal of OK and let a be a fixed generator of the maximal ideal p/pα of OK /pα . If R ⊆ OK /pα is a system of representatives for the field OK /p, then every element x ∈ OK /pα admits a unique representation in the form x = u0 + u1 a + · · · + uα−1 aα−1 , where ui ∈ R, for i = 1, . . . , α − 1. 2
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Proof. Let x be an element of OK /pα and let ψ : OK /pα → OK /p be the canonical epimorphism. Let u0 be the unique element in R such that ψ(u0 ) = ψ(x). Then, x − u0 lies in the kernel of ψ, that is, x − u0 ∈ a. Therefore, x = u0 + ya, for some y ∈ OK /pα . Now, assume that x = u 0 + y a, where u 0 ∈ R and y ∈ OK /pα . Then, both x − u0 and x − u 0 lie in the ideal a. Hence, u0 − u 0 also lies in a and so u0 = u 0 . Similarly y = u1 + za, for a unique u1 ∈ R and z ∈ OK /pα . Therefore, x = u0 + ya = u0 + u1 a + za2 . If we continue inductively with z and using the fact that aα = 0 we obtain the result. Lemma 2.2. Let OK be the ring of integers in a number field K, and let p be a prime ideal of OK . Let α and β be rational integers such that 0 ≤ β ≤ α, and let a be a fixed generator of the maximal ideal p/pα of OK /pα . Then, |U(OK /pα )aβ | = ϕK (pα−β ). Proof. By Lemma 2.1, if R is a system of representative for OK /p, then any element u ∈ U(OK /pα ) can be uniquely represented as u = u0 + u1 a + · · · + uα−1 aα−1 , where ui ∈ R, for i = 1, . . . , α − 1, and u0 0. Therefore, uaβ = u0 aβ + u1 aβ+1 + · · · + uα−β−1 aα−1 .
(6)
So, any element of U(OK /pα )aβ has a unique representation in the form (6). Now, define the function Φ : U(OK /pα )aβ → U(OK /pα−β ) by the formula Φ(u0 aβ + u1 aβ+1 + · · · + uα−β−1 aα−1 ) = u0 + u1 a + · · · + uα−β−1 aα−β−1 . It is clear that Φ is a bijection which completes the proof. Now, let us consider the general case. Let n = pα1 1 . . . pαs s , where p1 , . . . , p s are distinct nonzero prime ideals and α1 , . . . , α s are positive rational integers. Consider the action of the group of units U(OK /n) in the factor ring OK /n. From equation (3) it follows that to each element z ∈ OK /n corresponds an element (z1 , . . . , z s ) in OK /pα1 1 ⊕· · ·⊕OK /pαs s . If we denote by ai , for i = 1, . . . , s, a fixed generator of the maximal ideal pi /pi αi it follows that (z1 , . . . , z s ) = (u1 aβ11 , . . . , u s aβs s ), where ui is a unit in OK /pi αi and 0 ≤ βi ≤ αi . Therefore, the orbit U(OK /n)z decomposes as (U(OK /pα1 1 )aβ11 , . . . , U(OK /pαs s )aβs s ). Hence, from Lemma 2.2 and the fact that ϕK is multiplicative it follows that |U(OK /n)z| =
s i=1
|U(OK /pα1 i )aβi i | =
s
ϕK (pαi i −βi ) = ϕK (pα1 1 −β1 . . . pαs s −βs ).
i=1
The key tool to prove Theorem 1.1 will be the following result Lemma 2.3. Let OK be the ring of integers in a number field K, and let p be a prime ideal of OK . Let α and β be rational integers such that 0 ≤ β ≤ α. Let ψ : OK → OK /pα be the canonical epimorphism. Then, |{i ∈ OK /pα : ψ−1 (i) + pα = pβ }| = ϕK (pα−β ).
3
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Proof. As we have seen above the lattice of ideals of OK /pα is the chain 0 = aα ⊂ aα−1 ⊂ aα−2 ⊂ . . . ⊂ a ⊂ a0 = OK /pα , where a ∈ p/pα is a fixed generator of the maximal ideal p/pα . It is clear that for 0 ≤ β ≤ α we have ψ−1 (aβ ) = pβ and pβ + pα = pβ . On the other hand, for any element x ∈ p/pα we have x = aβ if and only if w(x) = β, that is, if and only if x = uaβ for a unit u. Now, the result follows from Lemma 2.2. We end this section with a lemma that relates the Euler totient function with the M¨obius function. For the ring Z of rational integers this result is well known, but for integers in number fields we have had trouble finding it in the literature. If n is a nonzero ideal of OK , then the analogue μK (n) of the classical M¨obius function is defined by μK (n) = 1, if n = OK ; μK (n) = (−1) s , if the prime decomposition of n is pα1 1 ...pαs s ,where α1 = α2 = ... = α s = 1 and μK (n) = 0 otherwise. Lemma 2.4. Let n = pα1 1 . . . pαs s . . . be the prime factorization of the ideal n of OK . Then, ϕK (n) = μK (pβ11 . . . pβs s )N(pα1 1 −β1 . . . pαs s −βs ).
(8)
0≤βi ≤αi
Proof. Write L(n) = 0≤βi ≤αi μK (pβ11 . . . pβs s )N(pα1 1 −β1 . . . pαs s −βs ). Since both the M¨obius function and the norm function are multiplicative it follows that L is multiplicative. Now ϕK is also multiplicative and consequently to prove the result it suffices to show that ϕK and L agree at the powers of nonzero prime ideals. Let p be a nonzero prime ideal and β a nonnegative rational integer. We have μK (pβ )N(pα−β ) = N(p)α − N(p)α−1 . L(pα ) = 0≤β≤α α
On the other hand, because OK /p is a local ring with maximal ideal p/pα we have ϕK (pα ) = |U(OK /pα )| = |OK /pα | − |p/pα | = N(p)α − N(p)α−1 , and therefore ϕK (n) = L(n) for every ideal n. 3. Proof of the main theorem We can now prove our main result. Proof of Theorem 1.1. First note that, if n = pα1 1 . . . pαs s is the prime factorization of the ideal n, then, as we have seen above, by the Chinese remainder theorem we have the isomorphism OK /n OK /pα1 1 ⊕ · · · ⊕ OK /pαs s . Hence, each element r ∈ OK /n can be identified with a k-tuple (r1 , ..., r s ), where ri ∈ OK /pαi i , for i = 1, ..., s, with componentwise addition and multiplication. Consequently, any ideal m/n of OK /n decomposes as m/n = pβ11 /pα1 1 ⊕ · · · ⊕ ps βs /pαs s , where 0 ≤ βi ≤ αi . From this we see that s α α β |{i ∈ OK /p j j : ψ−1 (i) + p j j = p j j }|. (9) |{i ∈ OK /n : ψ−1 (i) + n = m}| = j=1
Combining equation (9) with Lemma 2.3 and the fact that ϕK is multiplicative we obtain |{i ∈ OK /n : ψ−1 (i) + n = m| = ϕK (pα1 1 −β1 . . . pαs s −βs ). From this it follows that
i∈OK /n
f (ψ−1 (i) + n) =
f (pβ11 . . . pβs s )ϕK (pα1 1 −β1 . . . pαs s −βs ),
0≤βi ≤αi
as required. 4
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4. Applications to the generalized gcd-sum function For a positive rational integer n, the gcd-sum function, called also Pillai’s arithmetical function, is defined to be g(n) =
n
( j, n),
j=1
where ( j, n) denotes the greatest common divisor of i and n. This function arises in deriving asymptotic estimates for a lattice point counting problem (see section 5 of [4]). The gcd-sum function has been studied extensively in recent years by several authors (see e.g. [1, 3, 6, 10, 12, 17]). In [7] and [18] Pillai’s function has been extended to the ring of Gaussian integers. We now consider a generalization of the gcd-sum function to the ring of integers in a number field. If K is a number field and n is a nonzero ideal in the ring of integers OK that factors in primes as n = pα1 1 . . . pαs s , then we define the generalized gcd-sum function, denoted by gK , by the formula N(ψ−1 (i) + n), gK (n) = i∈OK /n
where ψ is the canonical epimorphism from OK onto OK /n. Theorem 1.1 gives N(pβ11 . . . pβs s )ϕK (pα1 1 −β1 . . . pαs s −βs ). gK (n) =
(10)
0≤βi ≤αi
Since both the norm and the generalized Euler totient function are multiplicative it follows that the generalized gcdsum function is multiplicative. Therefore, from the unique factorization in powers of prime ideals it follows that the generalized gcd-sum function is completely determined by its values at the powers of prime ideals. The following theorem gives the value of gK at powers of prime ideals. This generalizes to number fields Theorem 2.2 proved in [4]. Theorem 4.1. For every prime ideal p in the ring of integers OK of a number field K, and a positive rational integer α we have gK (pα ) = (α + 1)N(pα ) − αN(pα−1 ). Proof. In the case n = pα equation (10) becomes gK (pα ) =
N(pβ )ϕK (pα−β ).
0≤β≤α
For 0 ≤ β < α it follows from Lemma 2.4 that ϕK (pα−β ) = N(p)α−β − N(p)α−β−1 . Hence, N(pβ )(N(p)α−β − N(p)α−β−1 ) = (α + 1)N(pα ) − αN(pα−1 ), gK (pα ) = N(pα ) + 0≤β<α
as required.
5. Dirichlet series In [4] the author defines a Dirichlet series based on Pillai’s function and discuss some of its properties. In this section we extend this definition to the setting of number fields. We were unable to generalise to number fields all the properties established in [4]. In fact, we only show that the Dirichlet series for the Pillai’s function in Number Fields can be continued analytically to a meromorphic function. So, the problem of generalising to the setting of number fields all the the properties proved in [4] remains open. For a number field K and a complex variable s = σ + it the Dedekind zeta function of K is defined for R(s) > 1 by the series 1 , (11) ζK (s) = N(n) s n 5
C. Miguel / 00 (2017) 1–6
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where the sum ranges over all nonzero integral ideals n of OK . The Dedekind zeta-function can be continued analytically to a meromorphic function having a simple pole at s = 1; see [14, p.316] for details. For R(s) > 1, define the Dirichlet series based on Pillai’s function as G K (s) =
gK (n) . N(n) s n
The following theorem is an analog of [4, Theorem 4.1] for number fields. Theorem 5.1. The Dirichlet series for the Pillai function converges absolutely for R(s) > 2 and can be continued analytically to a meromorphic function having a double pole at s = 2 and a pole at every zero of ζK (s). Proof. Note that gK (n) =
N(a)ϕK (b).
ab=n
Hence, G K (s) =
⎞⎛ ⎞ ⎞ ⎛ ⎛ gK (n) ⎜⎜ N(n) ⎟⎟ ⎜⎜ ϕK (n) ⎟⎟ ϕK (n) ⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ = ζ (s − 1) ⎜⎜⎜⎜⎜ ⎟⎟⎟ . ⎜⎜⎜ = K ⎝ ⎝ s s⎠⎝ s⎠ s⎠ N(n) N(n) N(n) N(n) n n n n
Then, it follows from Lemma 2.4 that ⎞ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎜⎜⎜ μK (n) ⎟⎟⎟ ⎜⎜⎜ ϕK (n) ⎟⎟⎟ ⎜⎜⎜ μK (n) ⎟⎟⎟ ⎜⎜⎜ N(n) ⎟⎟⎟ 2 ⎟⎟ ⎜⎜ ⎟⎟ . ζK (s − 1) ⎜⎜⎝ ⎟⎟ = ζK (s − 1) ⎜⎜⎝ ⎟⎟ = ζK (s − 1) ⎜⎜⎝ N(n) s ⎠ N(n) s ⎠ ⎝ n N(n) s ⎠ N(n) s ⎠ n n n Finally, applying the identity
μK (n) n N(n) s
=
1 ζK (s)
(see [14, p.315]), the conclusion follows.
Acknowledgments The author expresses his gratitude to the anonymous reviewer for his/her helpful comments and suggestions, which have improved the paper. This work was supported by FCT project UID/EEA/50008/2013. [1] Abel, Ulrich; Awan, Waqar; Kushnirevych, Vitaliy A generalization of the gcd-sum function. J. Integer Seq. 16 (2013), no. 6, Article 13.6.7, 12 pp. [2] Atiyah, M. F.; Macdonald, I. G., Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969. [3] Bordell`es, Olivier, An asymptotic formula for short sums of gcd-sum functions. J. Integer Seq. 15 (2012), no. 6, Article 12.6.8, 5 pp. [4] Broughan, Kevin A., The gcd-sum function. J. Integer Seq. 4 (2001), no. 2, Article 01.2.2, 19 pp. ´ [5] Ces`aro, E., Etude moyenne du plus grand commun diviseur de deux nombre, Ann. Mat. Pura Appl. (1) 13 (1885), 235–250. [6] Chen, Shiqin; Zhai, Wenguang Reciprocals of the gcd-sum functions. J. Integer Seq. 14 (2011), no. 8, Article 11.8.3, 13 pp. [7] Dadayan, Zakhar Yu., The generalized S. S. Pillai function. (Russian) Ukr. Mat. Visn. 10 (2013), no. 1, 1–15, 144; translation in J. Math. Sci. (N. Y.) 192 (2013), no. 4, 377—388. [8] Dickson,L. E., History of the theory of numbers, vol. 1, Divisibility and primality, Chelsea Publ. Co. 1999. [9] Dummit, David S.; Foote, Richard M., Abstract algebra. Third edition. John Wiley & Sons, Inc., Hoboken, NJ, 2004. [10] Haukkanen, Pentti, On a gcd-sum function. Aequationes Math. 76 (2008), no. 1-2, 168—178. [11] Li, Yan; Kim, Daeyeoul, A Menon-type identity with many tuples of group of units in residually finite Dedekind domains. J. Number Theory 175 (2017), 42—50. [12] Liu, Huaning; Zhai, Wenguang, The equation n1 n2 = n3 n4 , the gcd-sum function and the mean values of certain character sums. Acta Arith. 152 (2012), no. 2, 137—157. [13] Miguel, C., Menon’s identity in residually finite Dedekind domains. J. Number Theory 137 (2014), 179—185. [14] Narkiewicz, W., Elementary and analytic theory of algebraic numbers. Third edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004. [15] Stewart, Ian; Tall, David, Algebraic number theory. Chapman and Hall Mathematics Series. Chapman and Hall, London; A Halsted Press Book, John Wiley & Sons, New York, 1979. [16] T´oth, L´aszl´o, A survey of gcd-sum functions. J. Integer Seq. 13 (2010), no. 8. [17] T´oth, L´aszl´o, Another generalization of the gcd-sum function. Arab. J. Math. (Springer) 2 (2013), no. 3, 313—320 [18] Varbanets, P. D.; Dadayan, Z. Yu., On the average value of a generalized Pillai function over Z[i] in the arithmetic progression. Ukrainian Math. J. 65 (2013), no. 6, 835—846.
6
C. Miguel
PII: DOI: Reference:
S0022-314X(17)30209-3 http://dx.doi.org/10.1016/j.jnt.2017.05.007 YJNTH 5772
To appear in:
Journal of Number Theory
Received date: Revised date: Accepted date:
1 February 2017 5 May 2017 6 May 2017
Please cite this article in press as: C. Miguel, Cesàro’s formula in number fields, J. Number Theory (2017), http://dx.doi.org/10.1016/j.jnt.2017.05.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
00 (2017) 1–6
Ces`aro’s formula in number fields C. Miguel Instituto de Telecomunica¸co˜ es, Beira Interior University, Department of Mathematics, Covilh˜a, Portugal, Email: [email protected]
Abstract We give an extension of a theorem of Ces`aro from the rational integers to the ring of integers of an arbitrary number field. This extension is used to generalize Pillai’s function to number fields. Keywords: Dedekind zeta function, Euler totient function, Gcd-sum function, Number field, Pillai function 2010 MSC: 11R42, 11R99
1. Introduction A theorem of Ces`aro (see e.g. [5], [8, p.127], [17]) states that for every natural number n ∈ N = {1, 2, . . .} and any arithmetical function f we have n f ((i, n)) = f (d)ϕ(n/d), (1) i=1
d|n
where (i, n) denotes the greatest common divisor of i and n, and ϕ is the Euler totient function. There is in the literature a large number of generalizations and analogues of Ces`aro theorem. For a rich and extensive survey concerning generalizations of Ces`aro theorem the reader is referred to [10]. Since rings of integers in a number field are natural generalizations of the rational integers, the question natural arises as to whether an analogous statement could be made for the ring of integers in a number field. Recently, there has been considerable interest in extending arithmetical identities from the rational integers to a more general setting (see e.g. [11], [13]). The aim of this paper is to extend Ces`aro theorem to the ring OK of integers in a number field K. Instead of working with elements in OK , where unique factorization can fail, we will work with ideals. As is well known using ideals in place of elements we can save unique factorization. More precisely, every nonzero ideal n of OK can be uniquely written in the form n = pα1 1 . . . pαs s , where p1 , . . . , p s are distinct nonzero prime ideals and α1 , . . . , α s are positive rational integers (see e.g. [14, p.8]). Unique factorization of ideals in OK permits calculations that are analogous to some familiar manipulations involving ordinary integers. In particular we can define the concept of arithmetical function on the set of ideals of OK . A real or complex-valued function defined on the set of ideals of the ring of integers in a number field is called an arithmetical function. As a very simple example, consider a nonzero ideal n of OK , then the generalized Euler totient function, which is denoted by ϕK (n), is defined to be the order of the multiplicative group of units in the factor ring OK /n, denoted by U(OK /n), with the convention that ϕK (OK ) = 1. That is, 1 i f n = OK , ϕK (n) = |U(OK /n)| otherwise. 1
C. Miguel / 00 (2017) 1–6
2
Note that, for each nonzero ideal n of OK , the residue class ring OK /n is finite (see e.g. [15, p.120]). Thus, ϕK is a finite valued function. Our main result is the following. Theorem 1.1. Let OK be the ring of integers in a number field K, and let n be a nonzero ideal of OK . Let ψ : OK → OK /n be the canonical epimorphism. Assume that n factorizes as n = pα1 1 . . . pαs s , where p1 , . . . , p s are distinct nonzero prime ideals and α1 , . . . , α s are positive rational integers. Then, for any arithmetical function f we have f (ψ−1 (i) + n) = f (pβ11 . . . pβs s )ϕK (pα1 1 −β1 . . . pαs s −βs ). (2) 0≤βi ≤αi
i∈OK /n
Note that, in view of the fact that OK /n is finite, the sum on the left is well-defined. By the unique factorization of ideals in OK it follows that the right side of equation (2) is equal to ab=n f (a)ϕK (b). 2. Some lemmas In this section we assemble the tools that we need to prove our results. Given a nonzero ideal n of OK , the group of units U(OK /n) acts naturally on the additive group of OK /n by left multiplication. Our next task is to describe and find the size of the orbits of this action. Let n = pα1 1 . . . pαs s be the prime factorization of the ideal n. By the Chinese remainder theorem for rings (see [9, p.265]) we obtain an isomorphism OK /n OK /pα1 1 ⊕ · · · ⊕ OK /pαs s .
(3)
U(OK /pαi i ),
Hence, the unit group U(OK /n) is the direct product of the groups for i = 1, . . . , s. In other words, the generalized Euler totient function ϕK is multiplicative. Consequently, we can limit ourselves to the case where the ideal n is a power of a nonzero prime ideal. So, let us assume that n = pα , where p is a nonzero prime ideal in OK and α is a positive rational integer. Note that, the ring OK /pα is a local ring with maximal ideal p/pα and all of its ideals are of the form pβ /pα , where 0 ≤ β ≤ α. Since every ideal in OK /pα is principal [2, p.99] it follows that the lattice of ideals of OK /pα is the chain 0 = aα ⊂ aα−1 ⊂ aα−2 ⊂ . . . ⊂ a ⊂ a0 = OK /pα , where a ∈ p/pα is a fixed generator of the maximal ideal p/pα . For a nonzero element x ∈ OK /pα let w(x) denote the largest integer m for which x ∈ am . Note that, if w(x) = m, then x may be uniquely represented in the form x = uam , where u is a unit in OK /pα . It is clear that the orbit of x is given by o(x) = U(OK /pα )aw(x) . Hence, the orbits of this action give rise to the following partition of OK /pα OK /pα = {0} ∪ U(OK /pα ) ∪ U(OK /pα )a ∪ U(OK /pα )a2 ∪ . . . ∪ U(OK /pα )aα−1 .
(4)
Our next goal is to find the size of each orbit. Note that p/pα is an additive subgroup of OK /pα of index N(p) = |OK /p|, the norm of the ideal p. Let C1 , . . . , C N(p) be the distinct cosets of p/pα in OK /pα . A set R = {u1 , . . . , uN(p) }, where ui ∈ Ci , for i = 1, . . . , N(p), is called a system of representatives for the field OK /p. Note that, if ψ : OK /pα → OK /p is the canonical epimorphism, then for each z ∈ OK /p there is a unique y ∈ R such that ψ(y) = z. Lemma 2.1. Let OK be the ring of integers in a number field K. Let p be a prime ideal of OK and let a be a fixed generator of the maximal ideal p/pα of OK /pα . If R ⊆ OK /pα is a system of representatives for the field OK /p, then every element x ∈ OK /pα admits a unique representation in the form x = u0 + u1 a + · · · + uα−1 aα−1 , where ui ∈ R, for i = 1, . . . , α − 1. 2
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Proof. Let x be an element of OK /pα and let ψ : OK /pα → OK /p be the canonical epimorphism. Let u0 be the unique element in R such that ψ(u0 ) = ψ(x). Then, x − u0 lies in the kernel of ψ, that is, x − u0 ∈ a. Therefore, x = u0 + ya, for some y ∈ OK /pα . Now, assume that x = u 0 + y a, where u 0 ∈ R and y ∈ OK /pα . Then, both x − u0 and x − u 0 lie in the ideal a. Hence, u0 − u 0 also lies in a and so u0 = u 0 . Similarly y = u1 + za, for a unique u1 ∈ R and z ∈ OK /pα . Therefore, x = u0 + ya = u0 + u1 a + za2 . If we continue inductively with z and using the fact that aα = 0 we obtain the result. Lemma 2.2. Let OK be the ring of integers in a number field K, and let p be a prime ideal of OK . Let α and β be rational integers such that 0 ≤ β ≤ α, and let a be a fixed generator of the maximal ideal p/pα of OK /pα . Then, |U(OK /pα )aβ | = ϕK (pα−β ). Proof. By Lemma 2.1, if R is a system of representative for OK /p, then any element u ∈ U(OK /pα ) can be uniquely represented as u = u0 + u1 a + · · · + uα−1 aα−1 , where ui ∈ R, for i = 1, . . . , α − 1, and u0 0. Therefore, uaβ = u0 aβ + u1 aβ+1 + · · · + uα−β−1 aα−1 .
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So, any element of U(OK /pα )aβ has a unique representation in the form (6). Now, define the function Φ : U(OK /pα )aβ → U(OK /pα−β ) by the formula Φ(u0 aβ + u1 aβ+1 + · · · + uα−β−1 aα−1 ) = u0 + u1 a + · · · + uα−β−1 aα−β−1 . It is clear that Φ is a bijection which completes the proof. Now, let us consider the general case. Let n = pα1 1 . . . pαs s , where p1 , . . . , p s are distinct nonzero prime ideals and α1 , . . . , α s are positive rational integers. Consider the action of the group of units U(OK /n) in the factor ring OK /n. From equation (3) it follows that to each element z ∈ OK /n corresponds an element (z1 , . . . , z s ) in OK /pα1 1 ⊕· · ·⊕OK /pαs s . If we denote by ai , for i = 1, . . . , s, a fixed generator of the maximal ideal pi /pi αi it follows that (z1 , . . . , z s ) = (u1 aβ11 , . . . , u s aβs s ), where ui is a unit in OK /pi αi and 0 ≤ βi ≤ αi . Therefore, the orbit U(OK /n)z decomposes as (U(OK /pα1 1 )aβ11 , . . . , U(OK /pαs s )aβs s ). Hence, from Lemma 2.2 and the fact that ϕK is multiplicative it follows that |U(OK /n)z| =
s i=1
|U(OK /pα1 i )aβi i | =
s
ϕK (pαi i −βi ) = ϕK (pα1 1 −β1 . . . pαs s −βs ).
i=1
The key tool to prove Theorem 1.1 will be the following result Lemma 2.3. Let OK be the ring of integers in a number field K, and let p be a prime ideal of OK . Let α and β be rational integers such that 0 ≤ β ≤ α. Let ψ : OK → OK /pα be the canonical epimorphism. Then, |{i ∈ OK /pα : ψ−1 (i) + pα = pβ }| = ϕK (pα−β ).
3
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Proof. As we have seen above the lattice of ideals of OK /pα is the chain 0 = aα ⊂ aα−1 ⊂ aα−2 ⊂ . . . ⊂ a ⊂ a0 = OK /pα , where a ∈ p/pα is a fixed generator of the maximal ideal p/pα . It is clear that for 0 ≤ β ≤ α we have ψ−1 (aβ ) = pβ and pβ + pα = pβ . On the other hand, for any element x ∈ p/pα we have x = aβ if and only if w(x) = β, that is, if and only if x = uaβ for a unit u. Now, the result follows from Lemma 2.2. We end this section with a lemma that relates the Euler totient function with the M¨obius function. For the ring Z of rational integers this result is well known, but for integers in number fields we have had trouble finding it in the literature. If n is a nonzero ideal of OK , then the analogue μK (n) of the classical M¨obius function is defined by μK (n) = 1, if n = OK ; μK (n) = (−1) s , if the prime decomposition of n is pα1 1 ...pαs s ,where α1 = α2 = ... = α s = 1 and μK (n) = 0 otherwise. Lemma 2.4. Let n = pα1 1 . . . pαs s . . . be the prime factorization of the ideal n of OK . Then, ϕK (n) = μK (pβ11 . . . pβs s )N(pα1 1 −β1 . . . pαs s −βs ).
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0≤βi ≤αi
Proof. Write L(n) = 0≤βi ≤αi μK (pβ11 . . . pβs s )N(pα1 1 −β1 . . . pαs s −βs ). Since both the M¨obius function and the norm function are multiplicative it follows that L is multiplicative. Now ϕK is also multiplicative and consequently to prove the result it suffices to show that ϕK and L agree at the powers of nonzero prime ideals. Let p be a nonzero prime ideal and β a nonnegative rational integer. We have μK (pβ )N(pα−β ) = N(p)α − N(p)α−1 . L(pα ) = 0≤β≤α α
On the other hand, because OK /p is a local ring with maximal ideal p/pα we have ϕK (pα ) = |U(OK /pα )| = |OK /pα | − |p/pα | = N(p)α − N(p)α−1 , and therefore ϕK (n) = L(n) for every ideal n. 3. Proof of the main theorem We can now prove our main result. Proof of Theorem 1.1. First note that, if n = pα1 1 . . . pαs s is the prime factorization of the ideal n, then, as we have seen above, by the Chinese remainder theorem we have the isomorphism OK /n OK /pα1 1 ⊕ · · · ⊕ OK /pαs s . Hence, each element r ∈ OK /n can be identified with a k-tuple (r1 , ..., r s ), where ri ∈ OK /pαi i , for i = 1, ..., s, with componentwise addition and multiplication. Consequently, any ideal m/n of OK /n decomposes as m/n = pβ11 /pα1 1 ⊕ · · · ⊕ ps βs /pαs s , where 0 ≤ βi ≤ αi . From this we see that s α α β |{i ∈ OK /p j j : ψ−1 (i) + p j j = p j j }|. (9) |{i ∈ OK /n : ψ−1 (i) + n = m}| = j=1
Combining equation (9) with Lemma 2.3 and the fact that ϕK is multiplicative we obtain |{i ∈ OK /n : ψ−1 (i) + n = m| = ϕK (pα1 1 −β1 . . . pαs s −βs ). From this it follows that
i∈OK /n
f (ψ−1 (i) + n) =
f (pβ11 . . . pβs s )ϕK (pα1 1 −β1 . . . pαs s −βs ),
0≤βi ≤αi
as required. 4
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4. Applications to the generalized gcd-sum function For a positive rational integer n, the gcd-sum function, called also Pillai’s arithmetical function, is defined to be g(n) =
n
( j, n),
j=1
where ( j, n) denotes the greatest common divisor of i and n. This function arises in deriving asymptotic estimates for a lattice point counting problem (see section 5 of [4]). The gcd-sum function has been studied extensively in recent years by several authors (see e.g. [1, 3, 6, 10, 12, 17]). In [7] and [18] Pillai’s function has been extended to the ring of Gaussian integers. We now consider a generalization of the gcd-sum function to the ring of integers in a number field. If K is a number field and n is a nonzero ideal in the ring of integers OK that factors in primes as n = pα1 1 . . . pαs s , then we define the generalized gcd-sum function, denoted by gK , by the formula N(ψ−1 (i) + n), gK (n) = i∈OK /n
where ψ is the canonical epimorphism from OK onto OK /n. Theorem 1.1 gives N(pβ11 . . . pβs s )ϕK (pα1 1 −β1 . . . pαs s −βs ). gK (n) =
(10)
0≤βi ≤αi
Since both the norm and the generalized Euler totient function are multiplicative it follows that the generalized gcdsum function is multiplicative. Therefore, from the unique factorization in powers of prime ideals it follows that the generalized gcd-sum function is completely determined by its values at the powers of prime ideals. The following theorem gives the value of gK at powers of prime ideals. This generalizes to number fields Theorem 2.2 proved in [4]. Theorem 4.1. For every prime ideal p in the ring of integers OK of a number field K, and a positive rational integer α we have gK (pα ) = (α + 1)N(pα ) − αN(pα−1 ). Proof. In the case n = pα equation (10) becomes gK (pα ) =
N(pβ )ϕK (pα−β ).
0≤β≤α
For 0 ≤ β < α it follows from Lemma 2.4 that ϕK (pα−β ) = N(p)α−β − N(p)α−β−1 . Hence, N(pβ )(N(p)α−β − N(p)α−β−1 ) = (α + 1)N(pα ) − αN(pα−1 ), gK (pα ) = N(pα ) + 0≤β<α
as required.
5. Dirichlet series In [4] the author defines a Dirichlet series based on Pillai’s function and discuss some of its properties. In this section we extend this definition to the setting of number fields. We were unable to generalise to number fields all the properties established in [4]. In fact, we only show that the Dirichlet series for the Pillai’s function in Number Fields can be continued analytically to a meromorphic function. So, the problem of generalising to the setting of number fields all the the properties proved in [4] remains open. For a number field K and a complex variable s = σ + it the Dedekind zeta function of K is defined for R(s) > 1 by the series 1 , (11) ζK (s) = N(n) s n 5
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where the sum ranges over all nonzero integral ideals n of OK . The Dedekind zeta-function can be continued analytically to a meromorphic function having a simple pole at s = 1; see [14, p.316] for details. For R(s) > 1, define the Dirichlet series based on Pillai’s function as G K (s) =
gK (n) . N(n) s n
The following theorem is an analog of [4, Theorem 4.1] for number fields. Theorem 5.1. The Dirichlet series for the Pillai function converges absolutely for R(s) > 2 and can be continued analytically to a meromorphic function having a double pole at s = 2 and a pole at every zero of ζK (s). Proof. Note that gK (n) =
N(a)ϕK (b).
ab=n
Hence, G K (s) =
⎞⎛ ⎞ ⎞ ⎛ ⎛ gK (n) ⎜⎜ N(n) ⎟⎟ ⎜⎜ ϕK (n) ⎟⎟ ϕK (n) ⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ = ζ (s − 1) ⎜⎜⎜⎜⎜ ⎟⎟⎟ . ⎜⎜⎜ = K ⎝ ⎝ s s⎠⎝ s⎠ s⎠ N(n) N(n) N(n) N(n) n n n n
Then, it follows from Lemma 2.4 that ⎞ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎜⎜⎜ μK (n) ⎟⎟⎟ ⎜⎜⎜ ϕK (n) ⎟⎟⎟ ⎜⎜⎜ μK (n) ⎟⎟⎟ ⎜⎜⎜ N(n) ⎟⎟⎟ 2 ⎟⎟ ⎜⎜ ⎟⎟ . ζK (s − 1) ⎜⎜⎝ ⎟⎟ = ζK (s − 1) ⎜⎜⎝ ⎟⎟ = ζK (s − 1) ⎜⎜⎝ N(n) s ⎠ N(n) s ⎠ ⎝ n N(n) s ⎠ N(n) s ⎠ n n n Finally, applying the identity
μK (n) n N(n) s
=
1 ζK (s)
(see [14, p.315]), the conclusion follows.
Acknowledgments The author expresses his gratitude to the anonymous reviewer for his/her helpful comments and suggestions, which have improved the paper. This work was supported by FCT project UID/EEA/50008/2013. [1] Abel, Ulrich; Awan, Waqar; Kushnirevych, Vitaliy A generalization of the gcd-sum function. J. Integer Seq. 16 (2013), no. 6, Article 13.6.7, 12 pp. [2] Atiyah, M. F.; Macdonald, I. G., Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969. [3] Bordell`es, Olivier, An asymptotic formula for short sums of gcd-sum functions. J. Integer Seq. 15 (2012), no. 6, Article 12.6.8, 5 pp. [4] Broughan, Kevin A., The gcd-sum function. J. Integer Seq. 4 (2001), no. 2, Article 01.2.2, 19 pp. ´ [5] Ces`aro, E., Etude moyenne du plus grand commun diviseur de deux nombre, Ann. Mat. Pura Appl. (1) 13 (1885), 235–250. [6] Chen, Shiqin; Zhai, Wenguang Reciprocals of the gcd-sum functions. J. Integer Seq. 14 (2011), no. 8, Article 11.8.3, 13 pp. [7] Dadayan, Zakhar Yu., The generalized S. S. Pillai function. (Russian) Ukr. Mat. Visn. 10 (2013), no. 1, 1–15, 144; translation in J. Math. Sci. (N. Y.) 192 (2013), no. 4, 377—388. [8] Dickson,L. E., History of the theory of numbers, vol. 1, Divisibility and primality, Chelsea Publ. Co. 1999. [9] Dummit, David S.; Foote, Richard M., Abstract algebra. Third edition. John Wiley & Sons, Inc., Hoboken, NJ, 2004. [10] Haukkanen, Pentti, On a gcd-sum function. Aequationes Math. 76 (2008), no. 1-2, 168—178. [11] Li, Yan; Kim, Daeyeoul, A Menon-type identity with many tuples of group of units in residually finite Dedekind domains. J. Number Theory 175 (2017), 42—50. [12] Liu, Huaning; Zhai, Wenguang, The equation n1 n2 = n3 n4 , the gcd-sum function and the mean values of certain character sums. Acta Arith. 152 (2012), no. 2, 137—157. [13] Miguel, C., Menon’s identity in residually finite Dedekind domains. J. Number Theory 137 (2014), 179—185. [14] Narkiewicz, W., Elementary and analytic theory of algebraic numbers. Third edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004. [15] Stewart, Ian; Tall, David, Algebraic number theory. Chapman and Hall Mathematics Series. Chapman and Hall, London; A Halsted Press Book, John Wiley & Sons, New York, 1979. [16] T´oth, L´aszl´o, A survey of gcd-sum functions. J. Integer Seq. 13 (2010), no. 8. [17] T´oth, L´aszl´o, Another generalization of the gcd-sum function. Arab. J. Math. (Springer) 2 (2013), no. 3, 313—320 [18] Varbanets, P. D.; Dadayan, Z. Yu., On the average value of a generalized Pillai function over Z[i] in the arithmetic progression. Ukrainian Math. J. 65 (2013), no. 6, 835—846.
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