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Arithmetic of generalized Dedekind sums and their modularity

  • Dohoon Choi , Byungheup Jun , Jungyun Lee EMAIL logo and Subong Lim
Published/Copyright: September 1, 2018
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Open Mathematics
From the journal Open Mathematics Volume 16 Issue 1

Abstract

Dedekind sums were introduced by Dedekind to study the transformation properties of Dedekind η function under the action of SL2(ℤ). In this paper, we study properties of generalized Dedekind sums si,j(p, q). We prove an asymptotic expansion of a function on ℚ defined in terms of generalized Dedekind sums by using its modular property. We also prove an equidistribution property of generalized Dedekind sums.

Keywords: Dedekind sum; Quantum modular form
MSC 2010: 11F20

1 Introduction

Dedekind sums are defined by

s(b,c):=∑h(modc)B¯1(hc)B¯1(bhc)(1)

for coprime integers b and c, where

B¯1(x):=x−[x]−12if x∈R∖Z,0if x∈Z.

These sums were introduced by Dedekind [1], and have been studied with applications in diverse areas of mathematics (for example, see [2,3,4]).

The Dedekind sum has been generalized and studied by many other authors from diverse standing (for example, see [5,6,7]). In this paper, we consider generalized Dedekind sums defined as follows. The Bernoulli polynomial Bi(x) is defined by the exponential generating function

∑i=0∞Bi(x)tii!=tetxet−1(2)

and Bi(x) denotes the periodic Bernoulli polynomial

B¯i(x):=Bi(〈x〉)for i≠1 or x∉Z,0for i=1 and x∈Z,(3)

where 〈x〉 := x – [x] and [x] denotes the greatest integer not exceeding x. Let i and j be nonnegative integers. Suppose that p is an integer and q is a positive integer with gcd(p, q) = 1. Then, generalized Dedekind sums are defined as

si,j(p,q):=∑k=1qB¯i(kq)B¯j(pkq)(4)

and the number i + j is called the weight of si,j(p, q). For convenience, we let

hi,j(p,q):=(−1)i+j1i!j!(si,j(p,q)−di,jBiBj),(5)

where Bn is the nth Bernoulli number and di,j is given by

di,j:=1if i=1 or j=1,0otherwise.

Let {e1, …, em} be the standard basis of ℚm. We define a vector FN(p, q) ∈ ℚN for an even positive integer N by

FN(p,q):=∑i=1Nfi(p,q)ei,(6)

where fi(p, q) = qN2−1p−N2+i−1hi−1,N+1−i(p,q) for i = 1, …, N. Then we define a function GN : ℚ → ℚN as

GN(x)=GN(pq):=FN(p,q).(7)

We remark that any rational number x can be uniquely written as pq for a positive integer q and an integer p which are relatively prime. Thus, the above mapping GN is well-defined, and has the following asymptotic expansion expressed in terms of Bernoulli numbers.

Theorem 1.1

We have an asymptotic expansion of the form

GN1n∼((−1)N2+1CN)GN(−n)+AN1n−1EN1n+2B12δN,2e2

as n → ∞, where

δi,j=0if i≠j,1if i=j,
AN(x) =ANpq:=qN2−i+1pi−N2(−1)i+jj−1N−i1≤i,j≤N,  CN:=(−1)j+1N−jN−i1≤i,j≤N,(8)

andEN(x) = ∑i=1Nei(x)eiis a vector-valued function defined by

ei(x)=eipq:=q1−ipi(−1)iNN−iBNN!+pi−NqN−i+1(−1)N−iNi−1BNN!+BN−iBi(N−i)!i!q(−1)i−1+BN−i+1Bi−1(N−i+1)!(i−1)!p(−1)i−1.(9)

Here, ab is defined by

ab=1if b=0,a(a−1)⋯(a−b+1)b!if b>0,(10)

for nonnegative integers a and b. Theorem 1.1 is proved by using the fact that GN can be understood as a modular object, which was recently introduced by Zagier [8].

Example 1.2

For a givenN, we can compute the asymptotic expansion ofGNexplicitly and it is written in terms of Bernoulli numbersBn. For example, ifN = 2, then

GN1n∼101−1GN(−n)+012nB2+B12+12nB2

asn → ∞. IfN = 4, then

GN1n∼−1000−3100−32−10−11−11GN(−n)+0124n2B4−16nB3B1+14B22−16nB3B1+124n2B4−124n2B4+14B22−13nB3B1+18n2B4124n2B4−16nB3B1+18n2B4

asn → ∞.

Dedekind sums also have a special property of distribution. For an even positive integer N and positive integers p, q which are coprime, we consider another vector-valued function

HN(p,q):=qN−2h1,N−1(p,q),…,hN−1,1(p,q)∈QN−1.(11)

Let

〈X〉:=〈x1〉,〈x2〉,…,〈xN−1〉∈[0,1)N−1

be the fractional part of the vector X = (x1, x2, … xN–1). The following theorem states that HN(p, q) satisfies the property of equidistribution.

Theorem 1.3

For an even positive integerN, there exists an integerRNsuch that the set of rational numbers

〈RNHN(p,q)〉|(p,q)=1,0<p<q(12)

is equidistributed in [0, 1)N–1.

Remark 1.4

Once for an integerRNthe set (12) is equidistributed in [0, 1)N–1, the same holds for any integer multiple ofRN. An example ofRNis given by

RN=N!βNrN,

whereβkis a positive integer such thatαkβkis the reduced fraction ofBk ≠ 0 and

rN:=lcmDenominator ofβNNi+1Bi+1BN−i−1|0≤i≤N−2.(13)

The following is the table for values ofRN forN = 2, 4, 6, 8, 10.

N246810
RN12720604803628800479001600.

The rest of the paper is organized as follows. Section 2 summarizes the properties of generalized Dedekind sums. In Section 3, we show that generalized Dedekind sums satisfy a modular property introduced by Zagier [8] and prove Theorem 1.1. In Section 4, we describe the distribution of generalized Dedekind sums and prove Theorem 1.3.

2 Reciprocity formulas

In this section, we prove the reciprocity formulas of generalized Dedekind sums, which can be induced from those of Dedekind-Rademacher sums.

2.1 Dedekind-Rademacher sums

In this subsection, we briefly review the definition and the reciprocity formulas of Dedekind-Rademacher sums based on the paper [6]. The following is the definition of Dedekind-Rademacher sums.

Definition 2.1

For a, b, c ∈ ℕ andx, y, z ∈ ℝ/ℤ, the Dedekind-Rademacher sum is defined by

Sm,nabcxyz:=∑h (mod c)B¯ma(h+z)c−xB¯nb(h+z)c−y.

Since reciprocity relations mix various pairs of indices (m, n), Hall, Wilson and Zagier [6] stated them in terms of the generating function

δabcxyzXYZ:=∑m,n≥01m!n!Sm,nabcxyzXam−1Ybn−1,

where X and Y are nonzero variables and the variable Z is defined as –X – Y. The following is the precise statement of the reciprocity formula of Dedekind-Rademacher sums.

Theorem 2.2

([6, [Section 4]). Leta, b, cbe three positive integers with no common factor, x, y, zthree real numbers, andX, Y, Zthree variables with sum zero. Then

δabcxyzXYZ+δbcayzxYZX+δcabzxyZXY=14if (x,y,z)∈(a,b,c)R+Z3,0otherwise.

2.2 Reciprocity formulas of generalized Dedekind sums

The following are the reciprocity formulas of generalized Dedekind sums, which are induced from Theorem 2.2.

Theorem 2.3

Letpandqbe positive integers with gcd(p, q) = 1. The generalized Dedekind sums satisfy the following reciprocity formulas.

  1. Fora, b ≥ 1 with odda + b,

    −qa−1∑i=0bha−1+i,b−i(p,q)(−p)ia−1+ia−1+pb−1∑j=0ahb−1+j,a−j(q,p)(−q)jb−1+jb−1=Ba−1Bb(a−1)!b!q(−1)b−1+BaBb−1a!(b−1)!p(−1)b−1.(14)

  2. Fora = 0 and odd b ≥ 1,

    hb−1,0(q,p)=h0,b−1(q,p)=p2−bBb−1(b−1)!.

Proof

  1. By the definition of Dedekind-Rademacher sums, we have the following equations

    Sm,n1pq000=∑h(modq)B¯mhqB¯nphq=sm,n(p,q),Sm,nq1p000=∑h(modq)B¯nhpB¯mqhp=sn,m(q,p),

    and

    Sm,npq1000=B¯m(0)B¯n(0).

    Then Theorem 2.2 implies that the sum

    ∑m,n≥01m!n!(sm,n(p,q)Xm−1Ypn−1+B¯m(0)B¯n(0)Ypm−1Zqn−1+sn,m(q,p)Zqm−1Xn−1)(15)

    is equal to 14.

    Suppose that m + n is even. Then, we have

    hm,n(p,q)=1m!n!(sm,n(p,q)+dm,nBmBn).

    Since Bk = 0 for odd integer k > 1 and B1 = −12, we see that

    hm,n(p,q)=1m!n!sm,n(p,q)−14em,n,

    where

    em,n:=1if n=m=1,0otherwise.

    Moreover, we have

    B¯m(0)B¯n(0)=BmBn−14em,n.

    Therefore, the following can be induced from the sum (15)

    ∑m,n≥0m+neven(hm,n(p,q)Xm−1Ypn−1+1m!n!BmBnYpm−1Zqn−1+hn,m(q,p)Zqm−1Xn−1)=0.(16)

    If we multiply 1pqXYZ on both sides of equation (16), then we obtain

    ∑m,n≥0m+neven(hm,n(p,q)XmYpnZq+1m!n!BmBnXYpmZqn+hn,m(q,p)XnZqmYp)=0.(17)

    From the relation X + Y + Z = 0, the equation (17) is rewritten as

     ∑m,n≥0m+nevenhm,n(p,q)(−1)m(Y+Z)mYpnZq+hn,m(q,p)(−1)nZqm(Y+Z)nYp=∑m,n≥0m+neven1m!n!BmBnYpmZqn(Y+Z).(18)

    By the binomial expansion of (Y + Z)n one can see that

    ∑m,n≥0m+neven∑k1=0mhm,n(p,q)(−1)mq−1p−nmk1Yk1+nZm−k1+1+∑m,n≥0m+neven∑k2=0nhn,m(q,p)(−1)nq−mp−1nk2Yn−k2+1Zk2+m=∑m,n≥0m+neven1m!n!BmBnYpmZqn(Y+Z).(19)

    Now, we compare the coefficients of YbZa on both sides of equation (19). If a, b ≥ 1, then we have

    ∑i=0bq−1p−(b−i)(−1)a+i−1ha+i−1,b−i(p,q)a+i−1i+∑j=0aq−(a−j)p−1(−1)b+j−1hb+j−1,a−j(q,p)b+j−1j=1(b−1)!a!Bb−1Bap−(b−1)q−a+1b!(a−1)!BbBa−1p−bq−(a−1).(20)

    This gives the first result after multiplying by qapb (–1)b–1 in both sides of equation (20).

  2. For the second result, suppose that a = 0 and b is an odd integer with b ≥ 1. Note that

    sb−1,0(p,q)=∑k=1qB¯b−1kqB¯0pkq=∑k=1qB¯b−1pkqB¯0p(pk)q=∑k=1qB¯b−1pkqB¯0kq  =s0,b−1(p,q).

    Here, we used the fact that B0(x) = 1 and gcd(p, q) = 1. If we compare the coefficients of Yb in both sides of equation (19), then we have

    hb−1,0(q,p)(−1)b−1p−1=1(b−1)!Bb−1p−(b−1),

    which gives the second result.□

The reciprocity formulas of generalized Dedekind sums can be expressed in terms of the vectors FN(p, q).

Corollary 2.4

Letpandqbe positive integers with gcd(p, q) = 1, andNan even positive integer. Then

ANpqFN(p,q)+BN(p,q)FN(p,q)=ENpq,

whereANpq, FN(p, q) andENpqare given as in (8), (6), and (9), respectively. Here, we defineBN(p, q) byBN(p, q) = (βi,j(p, q))1≤i,j≤Nand

βi,j(p,q)=qN2−i+1pi−N2(−1)i+jj−1i−1.(21)

Proof

Let N = a + b – 1 for a, b ≥ 1. By Theorem 2.3 (2), we have

−qa−1∑i=0bha−1+i,b−i(p,q)(−p)ia−1+ia−1=∑j=0b−1q−N2+apN2−a+1fa+j(p,q)(−1)j+1a−1+ja−1−qa−1pb(−1)bNa−1hN,0(p,q)=∑j=0b−1q−N2+apN2−a+1fa+j(p,q)(−1)j+1a−1+ja−1+qa−Npb(−1)b+1Na−1BNN!(22)

and

pb−1∑j=0ahb−1+j,a−j(q,p)(−q)jb−1+jb−1=∑k=0a−1p−N2+bqN2−b+1fb+k(q,p)(−1)kb−1+kb−1+pb−1qa(−1)aNb−1hN,0(q,p)=∑k=0a−1p−N2+bqN2−b+1fb+k(q,p)(−1)kb−1+kb−1+pb−Nqa(−1)aNb−1BNN!.(23)

If we change variables a ↦ N – i + 1, j ↦ j – a, b ↦ i, and k ↦ j – b in (22) and (23), then (14) can be written as

∑j=0NqN2−i+1pi−N2(−1)i+jj−1N−ifj(p,q)+∑j=0Npi−N2qN2−i+1(−1)i+jj−1i−1fj(q,p)=q1−ipi(−1)iNN−iBNN!+pi−NqN−i+1(−1)N−iNi−1BNN!+BN−iBi(N−i)!i!q(−1)i−1+BN−i+1Bi−1(N−i+1)!(i−1)!p(−1)i−1.

This gives the desired result.□

Besides the reciprocity formulas, generalized Dedekind sums satisfy the following properties.

Theorem 2.5

LetNbe an even positive integer, andi, jbe nonnegative integers with i + j = N. Then we have the following.

  1. hi,j(−p,q)=(−1)jhi,j(p,q)−2B12,if i=j=1;(−1)jhi,j(p,q),otherwise.

  2. hi,j(p + q, q) = hi,j(p, q).

Proof

  1. By the definition of si,j(p, q), we see that

    si,j(−p,q)=∑k=0q−1B¯ikqB¯j−pkq.

    It is known that Bj(1 – x) = (–1)jBj(x) for j ≥ 0. Therefore, we have

    B¯j−pkq=B¯j1−pkq=(−1)jB¯jpkq

    for j ≥ 0. From this, we obtain

    si,j(−p,q)=(−1)jsi,j(p,q).

    If i = j = 1, then we have

    hi,j(−p,q)=si,j(−p,q)−B12=−si,j(p,q)−B12=−hi,j(p,q)−2B12.

    Otherwise, we see that

    hi,j(−p,q)=(−1)i+j1i!j!si,j(−p,q)=(−1)j(−1)i+j1i!j!si,j(p,q)=(−1)jhi,j(p,q).

  2. Note that si,j(p + q, q) is equal to

    ∑k=0q−1B¯ikqB¯j(p+q)kq=∑k=0q−1B¯ikqB¯jpkq+k=∑k=0q−1B¯ikqB¯jpkq,

    which is si,j(p, q) by its definition. From this, we obtain the desired result that hi,j(p + q, q) = hi,j(p, q).□

3 Modular properties of generalized Dedekind sums

In this section, we show that the vector-valued function GN defined in (7) satisfies the modular property, which was introduced by Zagier [8].

3.1 Automorphic factor

To introduce a modular property for a vector-valued function, we need an automorphic factor for a vector-valued function.

Definition 3.1

An automorphic factor of rankmis a functionρ: SL2(ℤ)×ℍ → GLm(ℂ) satisfying the following conditions.

  1. The functionρsatisfies the cocycle relation

    ρ(γ1γ2,x)=ρ(γ1,γ2x)ρ(γ2,x)(24)

    forγ1, γ2 ∈ SL2(ℤ) and x ∈ ℍ, whereabcdx=ax+bcx+d.

  2. For a fixedγ ∈ SL2(ℤ), entries ofρ(γ, x) are rational functions ofxwith coefficients in ℚ.

For a fixed γ ∈ SL2(ℤ), the action of ρ(γ, x) on ℂm is defined by

ρ(γ,x)ej=∑i=1mρij(γ,x)ei,

where ρi,j(γ, x) is the (i, j)th entry of ρ(γ, x).

Let N be an even positive integer. Now, we define an automorphic factor ρN of rank N as follows. Let S:=0−110,T:=1101∈SL2(Z). We define

ρN(S,x):=((−1)N2+1CN)−1(25)

and

ρN(T,x):=DN(x)−1,(26)

where

CN:=(−1)jN−jN−i1≤i,j≤N

and

DN(x):=δi,jxx+1−N2+i−11≤i,j≤N.

Here, δi,j is the Kronecker delta. Then, it induces an automorphic factor of rank N.

Note that S and T generate SL2(ℤ). Therefore, from (25) and (26), we can compute ρN(γ) for any γ ∈ SL2(ℤ) by using the cocycle condition as in (24). To prove that ρN is an automorphic factor of rank N, it suffices to check that

ρN(S,Sx)ρN(S,x)=ρN(U,U2x)ρN(U,Ux)ρN(U,x)

and

(ρN(S,Sx)ρN(S,x))2=I,

where U := TS, ρN(U, x) = ρN(T, Sx)ρN(S, x) and I denotes the N × N identity matrix. To prove this, we need the following lemmas.

Lemma 3.2

LetNbe a positive integer. Letiandjbe integers with 1 ≤ i, j ≤ N.

  1. IfNis even and we let

    Λi,j:=∑k=1N(−1)k+j−1N−ki−1j−1N−k,

    then Λi,j = δi,j.

  2. ∑k=1NN−ki−1j−1k−1(−1)k−1=N−jN−i.

Proof

  1. Recall that

    ab=1if b=0,a(a−1)⋯(a−b+1)b!if b>0,(27)

    for nonnegative integers a and b. If we consider the binomial expansion of (1 –(1 – T))j–1, then we see that

    (1−(1−T))j−1=∑k=0N−1j−1k(−1)k(1−T)k.

    In the above equality, we used the fact that ab = 0 if a < b. Then we have

    (1−(1−T))j−1=∑k=1Nj−1N−k(−1)N−k(1−T)N−k=∑k=1Nj−1N−k(−1)N−k∑i=0N−1N−ki(−1)iTi=∑k=1Nj−1N−k(−1)N−k∑i=1NN−ki−1(−1)i−1Ti−1=∑k=1N∑i=1Nj−1N−kN−ki−1(−1)N−k+i−1Ti−1=∑i=1NΛi,j(−1)N+i+jTi−1.

    Since (1 –(1 – T))j–1 = Tj–1, we see that

    Λi,j(−1)N+i+j=0if i≠j,1if i=j.

    If i = j, then (–1)N+i+j = 1 since N is even. Therefore, we obtain the desired result that Λi,j = δi,j for 1 ≤ i, j ≤ N.

  2. We will use the induction on N. If N = 1, it is easy to see that (2) is true. Suppose that (2) is true for N ≥ 1. Let 1 < i < N + 1 and 1 ≤ j < N + 1. By the recursive formula, we have

    N+1−jN+1−i=N−jN−(i−1)+N−jN−i.

    Then, the induction hypothesis implies that

    N+1−jN+1−i=∑k=1NN−ki−2j−1k−1(−1)k−1+∑k=1NN−ki−1j−1k−1(−1)k−1.

    If we use the recursive formula again, then we have

    N+1−jN+1−i=∑k=1NN+1−ki−1j−1k−1(−1)k−1=∑k=1N+1N+1−ki−1j−1k−1(−1)k−1.

    The last equality follows from that N+1−ki−1=0i−1=0 if k=N+1 since i – 1 > 0.

    Now, we will check the remaining three cases: i = 1, i = N + 1, and j = N + 1. If i = 1 and 1 ≤ j ≤ N + 1, then we have

    ∑k=1N+1N+1−k0j−1k−1(−1)k−1=∑k=1N+1j−1k−1(−1)k−1=∑k=0Nj−1k(−1)k=δj,1=N+1−jN.

    If i = N + 1 and 1 ≤ j ≤ N + 1, then we obtain

    ∑k=1N+1N+1−kNj−1k−1(−1)k−1=NN=1=N+1−j0.

    Suppose that j = N + 1 and 2 ≤ i ≤ N. Then, we have

    ∑k=1N+1N+1−ki−1Nk−1(−1)k−1=∑k=1N+1Ni−1N+1−ik−1(−1)k−1=Ni−1∑k=1N+1N+1−ik−1(−1)k−1=Ni−1∑k=0NN+1−ik(−1)k=Ni−1(1−1)N+1−i=0=0N+1−i.

    Here, we used the identity

    N−(k−1)i−1Nk−1=Ni−1N−(i−1)k−1.

    □

Lemma 3.3

Letiandjbe integers with N ≥ j > i ≥1. Then

xjyi∑k,l=1Nzxkyzljkklli=(x+y+z)j−ij!i!(j−i)!.

Proof

If we apply the binomial expansion twice, then we obtain

(x+y+z)j−i=∑k=0j−i∑l=0kj−ikklxj−i−kylzk−l.

Then, one can see that

(x+y+z)j−i=∑k=ij∑l=ikj−ik−ik−il−ixj−kyl−izk−l=i!(j−i)!j!xjyi∑k=ij∑l=ik(zx)k(yz)ljkklli.

By the definition of ab as in (27), we obtain the desired result. □

Lemma 3.4

Suppose thatNis an even positive integer. Letiandjbe integers with 1 ≤ i, j ≤ N.

  1. ∑k=1NN−kN−iN−jN−k(−1)j+k=δi,j.

  2. Let

    Δi,j:=∑k,l=1Nx−1xN2+1−ixN2+1−l11−xN2+1−k(−1)l+k+jN−lN−iN−kN−lN−jN−k.

    Then

    Δi,j=(−1)N2+1δi,j.

Proof

  1. One can obtain (1) from Lemma 3.2 (1) by replacing i (resp. j) with N − i + 1 (resp. N − j + 1).

  2. Note that by the definition of (ab) as in (27), we have

    N−lN−iN−kN−lN−jN−k≠0

    only when i ≥ l ≥ k ≥ j. Therefore, if i < j, then Δi,j = 0. If i = j, then a nonzero term in the summation appears only when k = l = i = j. So, we have Δi,j=(−1)N2+1 . In the case of i > j, one can see that

    Δi,j=x−1xN2+1−ix−N2+111−x−N2+1(−1)j ×∑k,l=1N−11−xN−k(x−1)N−lN−lN−iN−kN−lN−jN−k.

    By Lemma 3.3, Δi,j is equal to

    (N−j)!(N−i)!(i−j)!x−1xN2+1−ixN2+1−i11−xN2+1−j(−1)j×((1−x)+x+(−1))i−j=0.

    This completes the proof. □

Now, we prove that ρN induces an automorphic factor of rank N satisfying the cocycle relation as in (24).

Proposition 3.5

LetρNbe defined by (25) and (26). Then

  1. ρN(S, Sx)ρN(S, x) = I,

  2. ρN(U, U2x)ρN(U, Ux)ρN(U, x) = I.

Proof

  1. Lemma 3.4 (1) implies that CN2 = I, and hence we have

    ρN(S,Sx)ρN(S,x)=I.

  2. For U = TS, we see that

    ρN(U,x)=(−1)N2+111−xN2+1−iN−jN−i(−1)j1≤i,j≤N, ρN(U,Ux)=(−1)N2+1xN2+1−iN−jN−i(−1)j1≤i,j≤N,

    and

    ρN(U,U2x)=(−1)N2+1x−1xN2+1−iN−jN−i(−1)j1≤i,j≤N.

    By Lemma 3.4 (2), we have

    ρN(U,U2x)ρN(U,Ux)ρN(U,x)=(−1)N2+1(Δi,j)1≤i,j≤N=I.

    This is the desired result. □

Remark 3.6

These kinds of automorphic factors are closely related with a monomial times a modular form. For example, we consider F(z) := z6Δ(z), whereΔ(z) is defined by

Δ(z)=q∏n=1∞(1−qn)24

andq = e2πizforz ∈ ℍ. Then, it is known thatΔ(z) is a modular form of weight 12 on SL2(ℤ). Note that

F(z+1)=(z+1)6z6F(z)

and

F−1z=F(z).

Hence, we see that F(z) is a modular form associated withρ, which is defined by

ρ(T,z)=z+1z6,ρ(S,z)=1.

3.2 Modular property of GN

With the automorphic factor ρN, we can state the transformation property of the vector-valued function GN. We define a slash operator associated with ρN as follows. Let f be a vector-valued function on ℚ, i.e., f is a sum of functions f=∑i=1mfiei , where fi is a function on ℚ for i = 1, …, m. Then we define

(f|ρNγ)(x):=∑i=1mfi(γx)ρN−1(γ,x)ei

for x ∈ ℚ and γ ∈ SL2(ℤ).

Lemma 3.7

The vector-valued functionGN : ℚ → ℂNsatisfies the functional equations

GN−GN|ρNS(x)=AN(x)−1EN(x)+2B12δN,2e2

for all positive rational numbersxand

GN−GN|ρNT(x)=0

for all x ∈ ℚ.

Proof

In this proof, we will use the properties of generalized Dedekind sums. For positive integers p and q which are relatively prime, by Corollary 2.4, we have

ANpqFN(p,q)+BN(p,q)FN(q,p)=ENpq,

where ANpq , BN(p, q), FN(p, q), and ENpq are defined as in (8), (21), (6), and (9), respectively.

Lemma 3.2 (1) implies that

ANpq−1=(vi,j(p,q))1≤i,j≤N,vi,j=−q−N2+j−1pN2−jN−ji−1

since (vi,j(p, q))1≤i,j≤N × ANpq = (Λi,j)1≤i,j≤N, which is the identity matrix. Lemma 3.2 (2) implies that

ANpq−1BN(p,q)=CN.

Therefore, we see that

GN(x)−(−CN)GN(1x)=AN(x)−1EN(x)

for any positive rational number x since GN(x) = FN(p, q), where x = pq . By Theorem 2.5 (1), we see that

GN(−x)=(−1)N2GN(x)−2B12e2if N=2,(−1)N2GN(x)if N≥4.

Therefore, we have the first desired result.

The second result directly comes from Theorem 2.5 (2) and the definition of ρN(T) as in (26). □

Note that AN(x)−1EN(x) is a rational function of p and q, where x = pq ∈ ℚ. But AN(x) and EN(x) are homogeneous of degree 1, and hence AN(x)−1EN(x) is homogeneous of degree 0. Therefore, AN(x)−1EN(x) is actually a rational function of x. This implies that this vector-valued function can be extended to ℝ ∖ {0}. The vector-valued function GN can be understood as a quantum modular form, which is a new modular object on ℚ introduced by Zagier [8]. Quantum modular forms were studied in connection with Maass forms, mock modular forms and Eichler integrals (for example, see [9, 10]).

3.3 Proof of Theorem 1.1

By Lemma 3.7, we have

GN(x)−((−1)N2+1CN)GN(−1x)=AN(x)−1EN(x)+2B12δN,2e2

for positive rational numbers x. If we let x = 1n for n ∈ ℕ, then

GN1n∼((−1)N2+1CN)GN(−n)+AN1n−1EN1n+2B12δN,2CNe2

as n → ∞.

3.4 Application of modular property to the arithmetic of Dedekind sums

With the cocycle property of ρ2, one can obtain an explicit expression of G2( pq ) in terms of the negative continued fraction of pq . For this, we recall the followings:

  1. G2pq=p−1h0,2(p,q)h1,1(p,q)=112p−1q−1s1,1(p,q)+14,

  2. ρ2(S,x)−1=−10−11,

  3. ρ2(T,x)−1=x+1x001,

  4. G2(x)−(G2|ρ2S)(x)=0B221x−3B12+B22x,

  5. G2(x) − (G2|ρ2T)(x) = 0.

By the cocycle condition as in (24), we obtain

G2(x)−(G2|ρ2TaS)(x)=G2(x)−ρ2−1(S,x)ρ2(Ta,Sx)−1G2(TaSx)=G2(x)−ρ2−1(S,x)G2(Sx)=0B221x−3B12+B22x(28)

for a positive integer a.

Now, we express pq by using the negative continued fraction as follows

pq=[a1,a2,…,an]:=a1−1a2−1⋯−1an

and define

piqi:=[ai,ai+1,…,an]

for 1 ≤ i ≤ n. Then, for 1 ≤ i ≤ n − 1, we see that

ai−110pi+1qi+1=piqi.(29)

Moreover, for 1 ≤ i ≤ n − 1, we also have

G2pi+1qi+1−ρ2−1ai−110,pi+1qi+1G2piqi=E2pi+1qi+1.(30)

For simplicity, we define

Fi:=ρ2−1ai−110,pi+1qi+1=piqi+10piqi+11.

By combining equations in (30), we have

G2pq=F1−1F2−1⋯Fn−1−1G2pnqn−E2p2q2−E2p3q3−⋯−E2pnqn.

Note that

F1−1F2−1⋯Fn−1−1=anpq0−anq′q1,

where

q′p′=[a2,a3,…,an−1].

Moreover, we have

G2pnqn=112an−13,G2pq=112p−1q−112s1,1(p,q)+3,

and

E2p2q2+E2p3q3+⋯+E2pnqn=0112−pq+a1+a2+⋯+an−n−14.

Finally, we obtain

12s1,1(p,q)=p−q′q−(a1+a2+⋯+an)+3(n−1).

Eventually, this formula gives a simple expression in 12 s1,1(p, q) modulo 1 that plays a crucial role of the proof for the equidistribution property of h1,1(x).

Remark 3.8

We note that in above examples, the generalized Dedekind sum s1,1(p, q) is completely determined by the initial values and transformation formulas byS=0−110  and  T=1101 . Along this line, restricting ourselves to the two dimensional case, the Apostol sum s1,n(p, q) is completely determined by a two term relation

(n+1)pqns1,n(p,q)+pnqs1,n(q,p)=∑i=0n−1n+1i(−1)iBiBn+1−ipiqn+1−i+nBn+2

together with the continued fraction ofpq(cf. [5]).

In this paper, we consider a similar property for generalized Dedekind sums si,j(p, q). Usually, generalized Dedekind sums si,j, (i ≠ 1, j ≠ 1) except the outermost case (i.e. Apostol sums) do not have two term reciprocity formula. Namely,

12!2!s2,2(p,q)−3p4!q3s0,4(q,p)+p3!q2s1,3(q,p)+p2!2!qs2,2(q,p)=f(p,q)

for a Laurent polynomial f(p, q). In this case, the reciprocity formula for s2,2(p, q) cannot be reduced to a Laurent polynomial, but involves a transcendental term s1,3(p, q).

However, if we consider a vector consisting of generalized Dedekind sums, then they have a two term reciprocity formula as follows

14!s0,4(p,q)13!s1,3(p,q)12!2!s2,2(p,q)13!s3,1(p,q)+−p3q3000−3p2q3p2q200−3pq32pq2pq0−1q31q2−1q114!s0,4(q,p)13!s1,3(q,p)12!2!s2,2(q,p)13!s3,1(q,p)=F(p,q)

for a column vector F(p, q) = (f1(p, q), …, f4(p, q))Tconsisting of Laurent polynomials inpandq. Thus, we are able to compute algorithmically the (vector of) generalized Dedekind sums of (p, q) of fixed weight using the continued fraction ofpq , once we obtain the reciprocity formula.

4 Distribution of generalized Dedekind sums

For an even positive integer N and positive integers p and q which are relatively prime, we consider a vector-valued function HN(p, q) defined in (11). In this section, we show that there exists an integer RN such that fractional parts of the vectors 〈RNHN(p, q)〉 are equidistributed. That is the image of these vectors under the projection ℝN onto ℝN/ℤN is equidistributed on the torus. There is a necessary and sufficient condition for this due to Weyl.

4.1 Weyl’s equidistribution criterion

We recall the statement of Weyl’s criterion on torus. For details, we refer to [11].

Theorem 4.1

(Weyl’s equidistribution criterion). A sequence

{sk=(s1(k),s2(k),…,sn(k))∈[0,1)n}k∈N

is equidistributed in [0, 1)nif and only if for everym = (m1, m2, …, mn) ∈ ℤn − {0},

limT→∞1T∑k=1Te(m⋅sk)=0,

wherem⋅sk=∑i=1nmisi(k)ande(x) denotes exp(2πix).

For a nonzero vector m ∈ ℤN−1 and a positive real number x, let E(m, x) be the average of the exponentials of (2πi) m ⋅ RNHN(p, q) defined by

E(m,x):=1#(p,q)|gcd(p,q)=1,p<q≤x∑0<q<x∑0<p<q(p,q)=1em⋅RNHNp,q.(31)

To apply Theorem 4.1, one needs to show that E(m, x) tends to 0 as x goes to ∞. This is done by relating an exponential sum to

∑0<p<q(p,q)=1em⋅RNHNp,q.(32)

4.2 Exponential sums of generalized Dedekind sums

We relate (32) to an exponential sum for a Laurent polynomial. Let us first recall the exponential sum for a Laurent polynomial F(x) ∈ ℤ[x, x−1].

Definition 4.2

For a positive integerqand F(x) ∈ ℤ[x, x−1], we define the exponential sum of modulusqof F(x) as

K(F,q):=∑x∈(Z/qZ)∗eqF(x),

whereeq(x):=exp2πixq.

Let FN(x) be the rank (N − 1) vector of Laurent polynomials

FN(x):=f1x,f2x,…,fN−1x,(33)

where fi(x)=αNrNN−1ix−i+N−1N−ixN−i . Here, αN denotes the numerator of BN, and rN is the integer defined as in (13). For a nonzero integer vector m of rank (N − 1), we have the following Laurent polynomial in x

m⋅FN(x)=αNrN∑i=1N−1miN−1ix−i+miN−1N−ixN−i.(34)

To prove the relation between the two exponential sums (32) and K(m ⋅ FN, q), we need the following theorem.

Theorem 4.3

[12, Theorem 1.1] LetNbe an even positive integer. For positive integersiandjwith i + j = N,

RNqN−2hij(p,q)−αNrN(p′)iN−1i+pjN−1jq∈Z,

where p′ is an integer such that p′p ≡ 1 (mod q), and RN = N! βNrNwithβN being the denominator ofBN.

By Theorem 4.3, one can see that

∑0<p<q(p,q)=1em⋅RNHNp,q=K(m⋅FN,q).

Therefore, we come to the estimation of the exponential sum of m ⋅ FN(x).

4.3 Bounds for exponential sums

Let q be a prime. Then, the estimation of K(m ⋅ FN, q), accompanied with some reductions, will be sufficient in showing Weyl’s criterion for

RNHN(p,q)|(p,q)=1,0<p<q

to be equidistributed. This will complete the proof of Theorem 1.3. To achieve the full estimation, we will follow the steps taken in [12].

The following lemma is necessary to prove that m ⋅ FN(x) has a Weil type bound for all but finitely many primes p.

Lemma 4.4

For positive integersiandj, letF(x)=∑k=−jiakxkbe a Laurent polynomial with integer coefficients such that ai ≠ 0 and a−j ≠ 0. Letpbe any prime withp ∤ ak for some k ≠ 0. Then

K(F,p)≤(i+j)p.

Proof

See Theorem 1.3 in [13]. □

Since m ⋅ FN is written as Laurent polynomial type in Lemma 4.4 (see equation (34)), we have the following.

Proposition 4.5

Letmbe a nonzero integer vector andm ⋅ FN(x) be the Laurent polynomial as give in (33). Putd = gcd(m1, m2, …, mN−1). Then for any positive integerp ∤ αNrNd, we have

K(m⋅FN,p)≤2(N−1)p.(35)

For a general modulus q, we have the following bound.

Proposition 4.6

Suppose thatqis a positive integer, and thatNis an even positive integer. Let DFN be a positive integer such that DFN||FN. Letω(q) be the number of prime factors ofq. Then

K(m⋅FN,q)≤DFN(12N−12)ω(q)q(1−13N−2).

From now on, we justify Proposition 4.6.

4.4 Reduction to prime modulus

If q has many prime factors, the bound is obtained by composing the bound previously obtained for primes dividing q. This is done by the next two reduction steps.

First, we consider the case q being a power of a prime p.

Lemma 4.7

Let F(x) be a Laurent polynomial with integer coefficients. Letpbe a fixed prime and pβ|| Then forα > β, we have

K(F,pα)=pβK(F~,pα−β),

whereF~(x)=1pβF(x).

Proof

We note that an element z ∈ (ℤ/pαℤ)∗ is written uniquely as z = pα−βx + y for x ∈ ℤ/pβℤ and y ∈ (ℤ/pα−βℤ)∗. Thus, we obtain

K(F,pα)=∑z∈(Z/pαZ)∗eFpα=∑x∈Z/pβZ∑y∈(Z/pα−βZ)∗eF~pα−β=pβK(F~,pα−β).

□

After the previous lemma, we can pull out p-factors out of the coefficients of m ⋅ FN(x). For positive integers i and j, let F(x)=∑k=−jiakxk be a Laurent polynomial with integer coefficients such that ai ≠ 0 and a−j ≠ 0. Let p be a prime such that F(x) ≢ 0 (mod p) and α be a positive integer.

Now, the following lemma is implied by Corollary 4.1 in [14] with the trivial character.

Lemma 4.8

With the above notation, suppose thatα ≥ 2 is an integer. Then

K(F,pα)≤4(i+2j)pα(1−1i+2j+1).

The previous two lemmas imply the following bound for prime powers.

Proposition 4.9

Letpbe a prime. Suppose thatmis any fixed nonzero vector in ℤN−1, and that pβ||m ⋅ FN(x) for some integerβ. Suppose thatα ≥ 2 is an integer such thatα > β. Then for anyp, we have

K(m⋅FN,pα)≤pβ12(N−1)pα(1−13N−2)≤DFN12(N−1)pα(1−13N−2),

where DFN is the largest positive integer such that DFN|FN.

Let us consider the case when q has several prime factors. We have the following effect of the Chinese remainder theorem for exponential sums.

Lemma 4.10

Let F(x) be a Laurent polynomial with integer coefficients, and let q1 > 1 and q2 > 1 be relatively prime integers. Let Fi(x) be the modqiChinese remainder of F(x) fori = 1, 2 (i.e. F ↦ (F1, F2) under the isomorphism (ℤ/q1q2ℤ)[x, x−1] → (ℤ/q1ℤ)[x, x−1] × (ℤ/q2ℤ)[x, x−1]). Then

K(F,q1q2)=K(F1,q1)K(F2,q2).

Proof

This is a consequence of Fubini theorem. □

4.5 Proof of Theorem 1.3

For x > 1, let ϕ(x) := |(ℤ/[x]ℤ)∗| be Euler’s phi function. By using ∑q<xϕ(q) ∽ x2 as x → ∞, we obtain the proof of the main theorem from Weyl’s criterion for equidistribution and Proposition 4.6.

Now, we apply Proposition 4.6 to deducing Weyl’s criterion from the bound of exponential sums. Note that ω(q) in Proposition 4.6 has a well-known estimation

ω(q)≤clog⁡qloglog⁡q(36)

for some constant c. For sufficiently large q,

(12N−12)ω(q)≤(12N−12)clog⁡qloglog⁡q≤(q)clog⁡(12N−12)loglog⁡q.

Thus, we obtain that for any ϵ > 0,

(12N−12)ω(q)≪qϵ.

Therefore, by Proposition 4.6, we have the following bound.

Proposition 4.11

LetNbe an even positive integer, and letmbe a nonzero integer vector of rank N − 1. Then

K(m⋅FN,q)≪q(1+ϵ−13N−2)

for allϵ > 0.

Weyl’s criterion for HN(p, q) comes from the following estimation

∑0<q<x∑0<p<q(p,q)=1em⋅RNHN(p,q)=∑0<q<xK(m⋅FN,q)≤x(2+ϵ−13N−2).(37)

Consequently, Weyl’s criterion is fulfilled for the fractional part of the vector HN(p, q):

E(m,x)=1#(p,q)|gcd(p,q)=1,p<q≤x∑0<q<x∑0<p<q(p,q)=1e(m⋅RNHN(p,q))→0(38)

as x → ∞. This completes the proof.

Acknowledgement

The authors would like to thank the referees for the valuable comments and helpful corrections, which improved the paper.

The second named author was supported by (NRF-2015R1D1A1A09059083) and the third named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and (NRF-2017R1A6A3A11030486) and the last author was supported by (NRF-2017R1C1B5017409).

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Received: 2018-02-05
Accepted: 2018-06-01
Published Online: 2018-09-01

© 2018 Choi et al., published by De Gruyter.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
https://doi.org/10.1515/math-2018-0082
Keywords for this article
Dedekind sum; Quantum modular form
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  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
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  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
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