On the Asymptotic Behavior of Dedekind Sums

Kurt Girstmair Institut für Mathematik Universität Innsbruck Technikerstr. 13/7 A-6020 Innsbruck Austria

               Kurt.Girstmair@uibk.ac.at

Abstract

Let z be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums S(p_{k}, q_{k}) belonging to convergents p_{k}/q_{k} of the regular continued fraction expansion of z with that of Dedekind sums S(s_{j}/t_{j}) belonging to convergents s_{j}/t_{j} of the negative regular continued fraction expansion of z. Whereas the three main cases of this behavior are closely related, a more detailed study of the most interesting case (in which the Dedekind sums remain bounded) exhibits some marked differences, since the cluster points depend on the respective periods of these expansions. We show in which cases cluster points of S(s_{j}, t_{j}) can coincide with cluster points of S(p_{k}, q_{k}). An important tool for our purpose is a criterion that says which convergents s_{j}/t_{j} of z are convergents p_{k}/q_{k}.

1 Introduction and results

Let z be a real irrational number. We consider the regular continued fraction expansion

z = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + ⋯}} = ⌊a_{0}, a_{1}, a_{2}, …⌋,

which is defined by the following algorithm:

z_{0} = z, a_{0} = ⌊z_{0}⌋, z_{j+1} = 1/(z_{j} − a_{j}), a_{j+1} = ⌊z_{j+1}⌋, j ≥ 0.

The convergents p_{0}/q_{0}, p_{1}/q_{1}, p_{2}/q_{2}, … of this expansion are given by

p_{−1} = 1, q_{−1} = 0, p_{0} = a_{0}, q_{0} = 1, p_{k+1} = a_{k+1}p_{k} + p_{k−1}, q_{k+1} = a_{k+1}q_{k} + q_{k−1}, k ≥ 0.

For the sake of simplicity we call these convergents the ⌊ ⌋-convergents of z.

We will compare the regular continued fraction expansion of z with the negative regular continued fraction expansion

z = c_{0} − \frac{1}{c_{1} − \frac{1}{c_{2} − ⋯}} = ⌈c_{0}, c_{1}, c_{2}, …⌉,

which is defined by the algorithm

z_{0} = z, c_{0} = ⌈z_{0}⌉, z_{j+1} = 1/(c_{j} − z_{j}), c_{j+1} = ⌈z_{j+1}⌉, j ≥ 0.

The convergents s_{0}/t_{0}, s_{1}/t_{1}, s_{2}/t_{2}, … of this expansion are given by

s_{−1} = 1, t_{−1} = 0, s_{0} = c_{0}, t_{0} = 1, s_{j+1} = c_{j+1}s_{j} − s_{j−1}, t_{j+1} = c_{j+1}t_{j} − t_{j−1}, j ≥ 0.

They are called the ⌈ ⌉-convergents of z. Note that c_{j} ≥ 2 for all j ≥ 1, and c_{j} ≥ 3 for infinitely many indices j ≥ 0 (see [8]). Henceforth we call this continued fraction expansion simply the negative regular expansion. This expansion has aroused some interest due to the work of Hirzebruch and Zagier about class numbers of quadratic fields (see [6], [7, p. 136]).

It is well-known that ⌊ ⌋-convergents of z have optimal approximation properties which ⌈ ⌉-convergents have not, unless they happen to be ⌊ ⌋-convergents (see [4, p. 44 ff.]). In general, ⌈ ⌉-convergents are only intercalary fractions of ⌊ ⌋-convergents: If

\frac{p_{k}}{q_{k}} and \frac{p_{k+1}}{q_{k+1}} = \frac{a_{k+1}p_{k} + p_{k−1}}{a_{k+1}q_{k} + q_{k−1}}

are two adjacent ⌊ ⌋-convergents of z, then the intercalary fractions are

\frac{p_{k} + p_{k−1}}{q_{k} + q_{k−1}} , \frac{2p_{k} + p_{k−1}}{2q_{k} + q_{k−1}} , …, \frac{(a_{k+1} − 1)p_{k} + p_{k−1}}{(a_{k+1} − 1)q_{k} + q_{k−1}} .

They also approximate z successively, but not as well as ⌊ ⌋-convergents do.

It is, therefore, desirable to be able to decide whether a ⌈ ⌉-convergent is a ⌊ ⌋-convergent. We give the following criterion.

Theorem 1. Let z = ⌈c_{0}, c_{1}, c_{2}, …⌉ be an irrational number and s_{j}/t_{j}, j ≥ 1, a ⌈ ⌉- convergent of z. Then s_{j}/t_{j} is a ⌊ ⌋-convergent of z iff c_{j+1} ≥ 3. In particular, the sequence s_{j}/t_{j}, j ≥ 1, contains infinitely many ⌊ ⌋-convergents of z.

Example 2. Let z = e = 2.71828 ⋯ be Euler’s number, whose regular continued fraction expansion is

⌊2, \overline{1, 2k, 1}⌋_{k=1}^{∞} = ⌊2, 1, 2, 1, 1, 4, 1, 1, 6, 1, …⌋,

where we have used Perron’s notation (see [4, p. 124]). By means of the transition formula

⌊a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, …⌋ = ⌈a_{0} + 1, 2^{(a_{1}−1)}, a_{2} + 2, 2^{(a_{3}−1)}, a_{4} + 2, …⌉  (1)

(see [1, p. 93]) one easily obtains

e = ⌈\overline{3, 4k, 3, 2^{(4k−1)}}⌉_{k=1}^{∞} = ⌈3, 4, 3, 2^{(3)}, 3, 8, 3, 2^{(7)}, …⌉;

here 2^{(j)} stands for a sequence of j numbers 2. According to Theorem 1, the ⌈ ⌉-convergents

\frac{s_{1}}{t_{1}} = \frac{11}{4}, \frac{s_{5}}{t_{5}} = \frac{87}{32}, \frac{s_{6}}{t_{6}} = \frac{193}{71}, \frac{s_{7}}{t_{7}} = \frac{1457}{536}, \frac{s_{15}}{t_{15}} = \frac{23225}{8544}, \frac{s_{16}}{t_{16}} = \frac{49171}{18089}, \frac{s_{17}}{t_{17}} = \frac{566827}{208524}

are ⌊ ⌋-convergents of e. Indeed, if the latter are denoted by p_{k}/q_{k}, k ≥ 0, we find that the above ⌈ ⌉-convergents coincide with p_{3}/q_{3}, p_{5}/q_{5}, …, p_{15}/q_{15}, respectively. As intercalary fractions we have, for instance, s_{2}/t_{2} = (p_{4} + p_{3})/(q_{4} + q_{3}) or s_{8}/t_{8} = (p_{10} + p_{9})/(q_{10} + q_{9}).

The main application of Theorem 1 in this paper is a comparative study of the asymptotic behavior of Dedekind sums with arguments near quadratic irrationals. For an integer a and a natural number b let

s(a, b) = ∑_{k=1}^{b}((k/b))((ak/b))

be the classical Dedekind sum, where

((x)) = \begin{cases}x − ⌊x⌋ − 1/2, & if x ∈ ℝ ∖ ℤ; \\ 0, & if x ∈ ℤ \end{cases}

(see, for instance, [5, p. 1]). We work with S(a, b) = 12s(a, b) instead of s(a, b) and compare the asymptotic behavior of the Dedekind sums S(p_{k}, q_{k}) of a quadratic irrational z with the asymptotic behavior of S(s_{j}, t_{j}) when k and j tend to infinity. Theorem 1 has the effect that the asymptotic behavior is, roughly speaking, the same in both cases.

For a quadratic irrational z both expansions are periodic, i.e.,

z = ⌊a_{0}, a_{1}, … a_{q}, \overline{b_{1}, b_{2}, …, b_{l}}⌋ = ⌈c_{0}, c_{1}, … c_{r}, \overline{d_{1}, …, d_{m}}⌉,

where (b_{1}, …, b_{l}) and (d_{1}, …, d_{m}) are the respective periods (see [8, Satz 14, Satz 15]). Here l and m are smallest possible. In addition, q and r are smallest possible for this choice of l and m. In the purely periodic case, we put q = −1 and r = −1. Further, let

L = \begin{cases}l, & if l is even; \\ 2l, & if l is odd. \end{cases}

Then (b_{1}, …, b_{L}) is also a period of the regular continued fraction expansion of z. If l is odd, it has the form (b_{1}, …, b_{l}, b_{1}, …, b_{l}). The three cases of Theorem 3 (below) are connected with the values of

B = (−1)^{q} ∑_{j=1}^{L}(−1)^{j−1}b_{j} and D = \frac{1}{m} ∑_{j=1}^{m} d_{j}.

It should be noted that B = 0 if l is odd or if the period (b_{1}, …, b_{l}) is palindromic.

Theorem 3. In the above setting, B > 0, B = 0, B < 0 iff D < 3, D = 3, D > 3, respectively. Suppose that j and k tend to ∞. Then

both S(p_{k}, q_{k}) and S(s_{j}, t_{j}) tend to ∞ if B > 0 (or D < 3),

these quantities remain bounded if B = 0 (or D = 3),

and they tend to −∞ if B < 0 (or D > 3).

In the remainder of this section we assume B = 0, i.e., D = 3. In the paper [2] it has been shown that the Dedekind sums S(p_{k}, q_{k}) accumulate near L cluster points U_{h}, h = 1, …, L. They are given by

U_{h} = A + (−1)^{q} ∑_{k=1}^{h}(−1)^{k−1}b_{k} + \begin{cases}z + 1/u_{h} − 3, & if q + h is odd ; \\ z − 1/u_{h}, & if q + h is even. \end{cases}  (2)

Here A = ∑_{k=0}^{q}(−1)^{k−1}a_{k} and u_{h} = ⌊\overline{b_{h}, b_{h−1}, …, b_{1}, b_{L}, b_{L−1}, …, b_{h+1}}⌋ is a purely periodic quadratic irrational, h = 1, …, L. On the other hand, we shall see that the Dedekind sums S(s_{j}, t_{j}) accumulate near cluster points V_{i}, i = 1, …, m. They are given by

V_{i} = C + ∑_{j=1}^{i}(3 − d_{j}) + z − 1/v_{i} − 3,  (3)

where C = ∑_{j=0}^{r}(3 − c_{j}) and v_{i} = ⌈\overline{d_{i}, d_{i−1}, …, d_{1}, d_{m}, d_{m−1}, …, d_{i+1}}⌉ is also a quadratic irrational, i = 1, …, m. In the purely periodic cases, we have A = 0 and C = 0.

Theorem 4. In the above setting, the cluster points U_{h}, h = 1, …, L, are pairwise distinct, as are the cluster points V_{i}, i = 1, …, m.

The following theorem concerns cluster points V_{i} which coincide with cluster points U_{h}.

Theorem 5. Let (d_{1}, …, d_{m}) be the period of the negative regular expansion of z. Let i ∈ {1, …, m} be such that d_{i+1} ≥ 3 (if i = m this means d_{1} ≥ 3). Then the cluster point V_{i} coincides with some cluster point U_{h} such that q + h is odd. A cluster point U_{h} with q + h even cannot coincide with a cluster point V_{i}.

Remark 6. One can show that the period obtained from (b_{1}, …, b_{L}) by means of the transition formula (1) is shortest possible among the possible periods of the negative regular expansion of z — but we abstain from doing this here. Accordingly, there are L/2 indices i ∈ {1, …, m} such that d_{i+1} ≥ 3, the corresponding values of d_{i+1} being either b_{1} + 2, b_{3} + 2, …, b_{L−1} + 2 or b_{2} + 2, b_{4} + 2, …, b_{L} + 2. Hence there are L/2 coincidences of cluster points V_{i} with cluster points U_{h}. This means that each cluster point U_{h}, q + h odd, coincides with such a V_{i}. So we have a fairly complete picture: The cluster points U_{h}, q + h odd, coincide with the cluster points V_{i}, d_{i+1} ≥ 3. Moreover, there are no coincidences of cluster points U_{h}, q + h even, and cluster points V_{i}.

Example 7. Let z = 1/\sqrt{53} = ⌊0, 7, \overline{3, 1, 1, 3, 14}⌋ = ⌈1, 2^{(6)}, \overline{5, 3, 2, 2, 16, 2, 2, 3, 5, 2^{(13)}}⌉. Since l = 5 is odd, the sequences S(p_{k}, q_{k}) and S(s_{j}, t_{j}) remain bounded. Because we have five (= L/2) entries ≥ 3 in the period of the negative regular expansion of z, we have five common cluster points, namely V_{1} = U_{2} = (636 + 60\sqrt{53})/371 = 2.89166 ⋯, V_{4} = U_{4} = (−159 + 54\sqrt{53})/53 = 4.41747 ⋯, V_{7} = U_{6} = (−2862 + 60\sqrt{53})/371 = −6.53690 ⋯, V_{8} = U_{8} = (−1749 + 57\sqrt{53})/212 = −6.29261 ⋯, and V_{22} = U_{10} = (477 + 57\sqrt{53})/212 = 4.20738 · · ·. As we expect, there are no further coincidences between the remaining 17 cluster point V_{i} and the remaining 5 cluster points U_{h}, although these points may lie close together, like V_{12} = (−13833 + 97\sqrt{53})/2332 = −5.62900 ⋯ and U_{7} = (−1749 − 46\sqrt{53})/371 = −5.61694 · · ·.

2 Proofs

Proof of Theorem 1. Let z_{j+1} denote the (j + 1)th complete quotient of the negative regular expansion of z. Accordingly,

z = \frac{z_{j+1}s_{j} − s_{j−1}}{z_{j+1}t_{j} − t_{j−1}} .

This gives

z − \frac{s_{j}}{t_{j}} = \frac{s_{j}t_{j−1} − s_{j−1}t_{j}}{t_{j}(z_{j+1}t_{j} − t_{j−1})}.

Here we use s_{j}t_{j−1} − s_{j−1}t_{j} = −1 (see [8, formula (4)]) and obtain

|z − \frac{s_{j}}{t_{j}} | = \frac{1}{t_{j}(z_{j+1}t_{j} − t_{j−1})} = \frac{1}{t_{j}^{2}} · \frac{t_{j}}{z_{j+1}t_{j} − t_{j−1}} .  (4)

We need the regular continued fraction expansion s_{j}/t_{j} = ⌊a_{0}, …, a_{n}⌋. However, we do not require a_{n} ≥ 2; hence we may assume that n is odd. Let p_{0}/q_{0}, …, p_{n}/q_{n} be the convergents of this expansion, in particular, s_{j}/t_{j} = p_{n}/q_{n}. Now we can apply a criterion of Legendre (see [4, p. 39]) to the identity (4); thereby, we see that s_{j}/t_{j} is a ⌊ ⌋-convergent of z, iff

\frac{t_{j}}{z_{j+1}t_{j} − t_{j−1}} < \frac{q_{n}}{q_{n} + q_{n−1}} .  (5)

Let s^{∗} ∈ {1, …, t_{j} − 1} denote the inverse of s_{j} mod t_{j}, i.e., s_{j}s^{∗} ≡ 1 mod t_{j} (observe j ≥ 1). Since s_{j}t_{j−1} − s_{j−1}t_{j} = −1, we see that t_{j−1} ≡ −s^{∗} mod t_{j}, and 1 ≤ t_{j−1} < t_{j} implies t_{j−1} = t_{j} − s^{∗}. On the other hand, s_{j} = p_{n} and t_{j} = q_{n}, which yields

s_{j}q_{n−1} − p_{n−1}t_{j} = (−1)^{n−1} = 1

(see [4, p. 25]). Accordingly, q_{n−1} = s^{∗}. Therefore, the condition (5) is equivalent to

z_{j+1}t_{j} − (t_{j} − s^{∗}) > t_{j} + s^{∗}, i.e., to z_{j+1} > 2.

Recall that z_{j+1} > 2 iff c_{j+1} ≥ 3. Recall, further, that c_{j+1} ≥ 3 holds for infinitely many indices j. Thereby, we obtain the desired result.

Proof of Theorem 3. Let z = ⌊a_{0}, a_{1}, … a_{q}, \overline{b_{1}, b_{2}, …, b_{l}}⌋ = ⌈c_{0}, c_{1}, … c_{r}, \overline{d_{1}, …, d_{m}}⌉ be as above. In [2] we studied the asymptotic behavior of S(p_{k}, q_{k}) for the ⌊ ⌋-convergents p_{k}/q_{k} of z. Indeed, if k = q + nL + h, n ≥ 0, h ∈ {1, …, L},

S(p_{k}, q_{k}) = A + nB + (−1)^{q} ∑_{j=1}^{h}(−1)^{j−1}b_{j} + \begin{cases}(p_{k} + q_{k−1})/q_{k} − 3, & if k is odd ; \\ (p_{k} − q_{k−1})/q_{k}, & if k is even. \end{cases} (6)

Here A and B are as in Section 1. Since p_{k}/q_{k} → z for k → ∞ and 0 ≤ q_{k−1}/q_{k} ≤ 1 for k ≥ 0, this gives

S(p_{k}, q_{k}) → ∞ if B > 0,
S(p_{k}, q_{k}) remains bounded if B = 0, and  (7)
S(p_{k}, q_{k}) → −∞ if B < 0.

A similar asymptotic behavior takes place for the Dedekind sums S(s_{j}, t_{j}) belonging to the ⌈ ⌉-convergents s_{j}/t_{j} of z. Indeed, a formula of Hirzebruch, Zagier and Myerson (see [3]) says

S(s_{j}, t_{j}) = ∑_{k=0}^{j}(3 − c_{k}) + (s_{j} − t_{j−1})/t_{j} − 3,

where we have written z = ⌈c_{0}, c_{1}, c_{2}, …⌋ disregarding the period. Hence we have, for j = r + nm + i, n ≥ 0, i ∈ {1, …, m},

S(s_{j}, t_{j}) = C + nm(3 − D) + ∑_{k=0}^{i}(3 − c_{k}) + (s_{j} − t_{j−1})/t_{j} − 3,  (8)

where C and D are as in Section 1. Observe s_{j}/t_{j} → z for j → ∞ and 0 ≤ t_{j−1}/t_{j} ≤ 1 for j ≥ 0 (see [8, Satz 4, Satz 1]). Then we obtain

S(s_{j}, t_{j}) → ∞ if D < 3,
S(s_{j}, t_{j}) remains bounded if D = 3, and  (9)
S(s_{j}, t_{j}) → −∞ if D > 3.

By Theorem 1, the sequence s_{j}/t_{j} contains infinitely many ⌊ ⌋-convergents p_{k}/q_{k} of z. Therefore, the asymptotic behavior of S(p_{k}, q_{k}) for these convergents of z in the sense of (7) must be the same as in the sense of (9). This, however, yields Theorem 3, in particular, the connection between B and D.

Remark 8. The relation between the quantities B and D can also be proved in a simple and direct way by means of the transition formula (1).

Proof of Theorem 4. In the paper [2] we deduced formula (2) from (6). In exactly the same way one can deduce (3) from (8). First we show that the cluster points V_{i}, i = 1, …, m, are pairwise distinct. Suppose that V_{i} = V_{j} for some i, j ∈ {1, …, m}. This implies

1/v_{i} = 1/v_{j} + a, for some a ∈ ℤ.

Since v_{i}, v_{j} are both > 1, we have 0 < 1/v_{i}, 1/v_{j} < 1, whence a = 0 and v_{i} = v_{j} follows. Since m is smallest possible, v_{i} and v_{j} have different negative regular expansions if i ≠ j. Therefore, i = j.

In the case of the cluster points U_{h}, h = 1, …, L, the proof is slightly more subtle. Suppose, first, that l is odd. Then the values 1/u_{h}, h = 1, 3, …, l, and 1/u_{l+h} = 1/u_{h}, h = 2, 4, …, l − 1, appear with the same sign in (2), whereas 1/u_{h}, h = 2, 4, …, l − 1, and 1/u_{l+h} = 1/u_{h}, h = 1, 3, …, l, appear with the opposite sign. If 1/u_{h} and 1/u_{k} have the same sign, one can argue as in the case of the cluster points V_{i}, since u_{h}, u_{k} are both > 1. So we are left with the case

1/u_{h} = −1/u_{k} + a, a ∈ ℤ.

Since 0 < 1/u_{h}, 1/u_{k} < 1, this is only possible with a = 1. Accordingly, we obtain

u_{h} = (1 − 1/u_{k})^{−1} = 1 + 1/(u_{k} − 1).

Now u_{k} = ⌊\overline{b_{k}, b_{k−1}, …, b_{1}, b_{l}, b_{l−1}, …, b_{k+1}}⌋. If b_{k} > 1, we obtain

1 + 1/(u_{k} − 1) = ⌊1, b_{k} − 1, \overline{b_{k−1}, …, b_{1}, b_{l}, …, b_{k}}⌋ = u_{h} = ⌊\overline{b_{h}, …, b_{1}, b_{l}, …, b_{h+1}}⌋.

However, u_{h} is purely periodic with period length l, and so must be 1 + 1/(u_{k} − 1). But if we compare the second entry b_{k} − 1 with the second entry that follows the period, we get the contradiction b_{k} − 1 = b_{k}. If b_{k} = 1, we obtain

1 + 1/(u_{k} − 1) = ⌊b_{k−1} + 1, \overline{b_{k−2}, …, b_{1}, b_{l}, …, b_{k−1}}⌋.

Again, 1+1/(u_{k} −1) must be purely periodic with period length l, which gives the impossible relation b_{k−1} + 1 = b_{k−1}.

If l = L is even, we proceed in a similar way: We have the same sign in (2) for 1/u_{h}, h = 1, 3, …, L − 1, and the opposite sign for 1/u_{h}, h = 2, 4, …, L and, thus, can rule out the corresponding identities if the signs are equal. In the case 1/u_{h} = −1/u_{k} + a, a ∈ ℤ, we argue as above.

Proof of Theorem 5. Let h ∈ {1, …, L}. We say that the ⌊ ⌋-convergent p_{k}/q_{k} belongs to the ⌊ ⌋-class h, if k = q + nL + h for some n ≥ 0. For the ⌊ ⌋-convergents p_{k}/q_{k} belonging to this class, the corresponding Dedekind sums S(p_{k}, q_{k}) converge against the cluster point U_{h} — as we have shown in the paper [2]. In the same way we say that the ⌈ ⌉-convergent s_{j}/t_{j} belongs to the ⌈ ⌉-class i, i ∈ {1, …, m}, if j = r + nm + i for some n ≥ 0. For ⌈ ⌉- convergents s_{j}/t_{j} belonging to this class, the corresponding Dedekind sums S(s_{j}, t_{j}) converge against the cluster point V_{i}.

Let i ∈ {1, …, m} be such that d_{i+1} ≥ 3 (if i = m, d_{1} ≥ 3). By Theorem 1, each convergent s_{j}/t_{j} of the ⌈ ⌉-class i equals some ⌊ ⌋-convergent p_{k}/q_{k}. Since s_{j}/t_{j} > z, k must be odd. Up to finitely many exceptions, these convergents s_{j}/t_{j} must belong to exactly one ⌊ ⌋-class h — otherwise the convergent sequence S(s_{j}, t_{j}) would have more than one cluster point. Accordingly, S(s_{j}, t_{j}) converges against U_{h} = V_{i} for this h. Because q + h ≡ k mod 2 for all p_{k}/q_{k} in the ⌊ ⌋-class h, we see that q + h is odd.

Finally, suppose that q + h is even and U_{h} = V_{i} for some i ∈ {1, …, m}. By (2) and (3), −1/u_{h} = −1/v_{i} + a, a ∈ ℤ. As in the proof of Theorem 4, we conclude u_{h} = v_{i}. Now

u_{h} = ⌊\overline{b_{h}, b_{h−1}, …, b_{1}, b_{L}, … b_{h+1}}⌋ = ⌈b_{h} + 1, \overline{2^{(b_{h−1}−1)}, b_{h−2} + 2, …, 2^{(b_{h+1}−1)}, b_{h} + 2}⌉.

Since v_{i} = ⌈\overline{d_{i}, d_{i−1}, …, d_{1}, d_{m}, … d_{i+1}}⌉ is purely periodic, we see that b_{h} + 1 must be equal to b_{h} + 2, which is impossible.

References

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[3] G. Myerson, On semi-regular continued fractions, Arch. Math. 48 (1987), 420–425.

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2010 Mathematics Subject Classification: Primary 11F20; Secondary 11A55. Keywords: Asymptotic behavior of Dedekind sums, continued fraction expansion of a quadratic irrational.

Received March 14 2014; revised version received July 3 2014. Published in Journal of Integer Sequences, July 11 2014.

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