Dedekind Sums with Arguments Near Euler’s Number e

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

                   Kurt.Girstmair@uibk.ac.at

Abstract We study the asymptotic behaviour of the classical Dedekind sums s(m/n) for convergents m/n of e, e^{2}, and (e+1)/(e−1), where e = 2.71828 … is Euler’s number. Our main tool is the Barkan-Hickerson-Knuth formula, which yields a precise description of what happens in all cases.

1 Introduction and results

Dedekind sums have quite a number of interesting applications in analytic number theory (modular forms), algebraic number theory (class numbers), lattice point problems and algebraic geometry (for instance [1, 7, 9, 12]).

Let n be a positive integer and m ∈ ℤ, (m, n) = 1. The classical Dedekind sum s(m/n) is defined by

s(m/n) = ∑_{k=1}^{n}((k/n))((mk/n))

where ((…)) is the usual sawtooth function (for example, [9, p. 1]). In the present setting it is more natural to work with

S(m/n) = 12s(m/n)

instead.

In the previous paper [3] we used the Barkan-Hickerson-Knuth-formula to study the asymptotic behaviour of S(s_{k}/t_{k}) for the convergents s_{k}/t_{k} of a periodic simple continued fraction α = [a_{0}, a_{1}, a_{2}, …], i. e., for a quadratic irrational α. In this situation two cases are possible: The sequence S(s_{k}/t_{k}) either remains bounded with a finite number of cluster points or it essentially behaves like C · k for some constant C depending on α. In the latter case S(s_{k}/t_{k}) − C · k remains bounded with finitely many cluster points. The former case occurs, for instance, if the period length of α is odd.

Since the order of magnitude of |S(m/n)| is log^{2} n on average [4], quadratic irrationalities produce Dedekind sums of a considerably smaller size. In fact, the inequality k ≤ 2 log t_{k}/ log 2 + 1 was already proved in 1841 [11]. Accordingly, if |S(s_{k}/t_{k})| is not bounded, we have |S(s_{k}/t_{k})| = O(log t_{k}) for a quadratic irrational α.

Because the structure of the continued fraction expansions of transcendental numbers like e or e^{2} is similar to that of quadratic irrationals [8, p. 123 ff.], nothing prevents us from applying the Barkan-Hickerson-Knuth-formula ((2) below) to these cases. It turns out that the asymptotic behaviour of Dedekind sums is quite similar to the case of quadratic irrationals. Only the case “S(s_{k}/t_{k}) bounded” cannot occur, as the said formula shows, since the continued fraction expansions of these numbers have unbounded digits. We shall show

Theorem 1. For a nonnegative integer k put

L(k) = \begin{cases}\frac{k}{3}, & if k ≡ 0, 1, 5 (mod 6); \\ −\frac{k}{3}, & otherwise. \end{cases}

Then we have, for the convergents s_{k}/t_{k} of Euler’s number e,

S(s_{k}/t_{k}) − L(k) = O (\frac{1}{k}) + \begin{cases}e − 3, & if k ≡ 3 (mod 6); \\ e − 3 − \frac{5}{6}, & if k ≡ 1 (mod 6); \\ e − 3 + \frac{2}{3}, & if k ≡ 2 (mod 6); \\ e − 3 + \frac{5}{6}, & if k ≡ 4 (mod 6); \\ e − 3 − \frac{2}{3}, & if k ≡ 5 (mod 6). \end{cases}

The continued fraction expansion of e^{2} = 7.38905 … is more complicated than that of e. This has the effect that the analogue of Theorem 1 also looks more complicated. We obtain

Theorem 2. For a nonnegative integer k put

L(k) = \begin{cases}−\frac{3k}{5}, & if k ≡ 1, 2, 3 (mod 10); \\ \frac{3k}{5}, & if k ≡ 6, 7, 8 (mod 10); \\ −\frac{6k}{5}, & if k ≡ 0, 4 (mod 10); \\ \frac{6k}{5}, & if k ≡ 5, 9 (mod 10). \end{cases}

Then we have, for the convergents s_{k}/t_{k} of the number e^{2},

S(s_{k}/t_{k}) − L(k) = O (\frac{1}{k}) + \begin{cases}e^{2} − 7, & if k ≡ 0 (mod 10); \\ e^{2} − \frac{37}{5}, & if k ≡ 1 (mod 10); \\ e^{2} − \frac{29}{5}, & if k ≡ 2 (mod 10); \\ e^{2} − \frac{31}{5} + \frac{1}{2}, & if k ≡ 3 (mod 10); \\ e^{2} − \frac{16}{5}, & if k ≡ 4 (mod 10); \\ e^{2} + 1, & if k ≡ 5 (mod 10); \\ e^{2} + \frac{7}{5}, & if k ≡ 6 (mod 10); \\ e^{2} − \frac{1}{5}, & if k ≡ 7 (mod 10); \\ e^{2} − \frac{4}{5} + \frac{1}{2}, & if k ≡ 8 (mod 10); \\ e^{2} − \frac{14}{5}, & if k ≡ 9 (mod 10). \end{cases}

Finally, we consider the case of e^{∗} = (e + 1)/(e − 1), which is fairly simple.

Theorem 3. For a nonnegative integer k put

L(k) = \begin{cases}−2k, & if k is even; \\ 2k, & if k is odd. \end{cases}

Then we have, for the convergents s_{k}/t_{k} of e^{∗},

S(s_{k}/t_{k}) − L(k) = O (\frac{1}{k}) + \begin{cases}e^{∗} − 2, & if k is even; \\ e^{∗} − 1, & if k is odd. \end{cases}

2 Proofs

We start with the continued fraction expansion [a_{0}, a_{1}, a_{2}, …] of an arbitrary irrational number. The numerators and denominators of its convergents s_{k}/t_{k} are defined by the recursion formulas

s_{−2} = 0, s_{−1} = 1, s_{k} = a_{k}s_{k−1} + s_{k−2} and
t_{−2} = 1, t_{−1} = 0, t_{k} = a_{k}t_{k−1} + t_{k−2}, for k ≥ 0.  (1)

The Barkan-Hickerson-Knuth formula says that for k ≥ 0

S(s_{k}/t_{k}) = ∑_{j=1}^{k}(−1)^{j−1}a_{j} + \begin{cases}(s′_{k} + t′_{k−1})/t′_{k} − 3, & if k is odd; \\ (s′_{k} − t′_{k−1})/t′_{k}, & if k is even; \end{cases}  (2)

[2], [5], [6]. Here s′_{k} and t′_{k} are defined as in (1), but for the number [0, a_{1}, a_{2}, …] instead of [a_{0}, a_{1}, a_{2}, …]. We prove the simplest case first.

Proof of Theorem 3. The digits a_{j} of the continued fraction expansion of e^{∗} are a_{j} = 4j + 2, j = 0, 1, 2, … [8, p. 124]. An easy calculation shows that for k ≥ 0

∑_{j=1}^{k}(−1)^{j−1}a_{j} = \begin{cases}−2k, & if k is even; \\ 2k + 4, & if k is odd. \end{cases}  (3)

Now s′_{k}/t′_{k} converges against [0, a_{1}, a_{2}, …] = e^{∗} − 2, and |e^{∗} − 2 − s′_{k}/t′_{k}| < 1/t_{k}^{′2} [8, p. 37]. We remarked in the Introduction that k = O(log t′_{k}). Hence we also have |e^{∗}−2−s′_{k}/t′_{k}| = O(1/k). Finally, (1) gives t′_{k−1}/t′_{k} = t′_{k−1}/(a_{k}t′_{k−1} + t′_{k−2}) ≤ 1/a_{k} = O(1/k). These observations, together with (2) and (3), prove the theorem.

Proof of Theorem 1. In the case of e = [a_{0}, a_{1}, a_{2}, …] one easily derives from [8, p. 124] that

a_{j} = \begin{cases}2, & if j = 0; \\ 2(j − 1)/3 + 2, & if j ≡ 2 (mod 3); \\ 1, & otherwise. \end{cases}

An elementary computation with arithmetic series (which is more laborious than that of the proof of Theorem 3) yields

∑_{j=1}^{k}(−1)^{j−1}a_{j} = \begin{cases}\frac{k}{3}, & if k ≡ 0 (mod 6); \\ −\frac{k}{3} + 1, & if k ≡ 3 (mod 6); \\ \frac{k−1}{3} + 1, & if k ≡ 1 (mod 6); \\ −\frac{k−1}{3}, & if k ≡ 4 (mod 6); \\ −\frac{k−2}{3} − 1, & if k ≡ 2 (mod 6); \\ \frac{k−2}{3} + 2, & if k ≡ 5 (mod 6). \end{cases}  (4)

In the same way as in the proof Theorem 3 we have s′_{k}/t′_{k} → e−2 and |e−2−s′_{k}/t′_{k}| = O(1/k). If k ≡ 2 (mod 3), we note t′_{k−1}/t′_{k} ≤ 1/a_{k} = O(1/k). If k ≡ 0 (mod 3) and k ≥ 3, we have

\frac{t′_{k−1}}{t′_{k}} = \frac{t′_{k−1}}{t′_{k−1} − t′_{k−2}} = \frac{1}{1 + t′_{k−2}/t′_{k−1}} .  (5)

Since t′_{k−2}/t′_{k−1} = O(1/k), this shows t′_{k−1}/t′_{k} = 1 + O(1/k). If k ≡ 1 (mod 3) and k ≥ 4, formula (5) also holds. Together with t′_{k−2}/t′_{k−1} = 1+O(1/k), it gives t′_{k−1}/t′_{k} = 1/2+O(1/k). These observations, combined with (2) and (4), prove the theorem.

Proof of Theorem 2. The proof follows the above pattern. One obtains from [8, p. 125]

a_{j} = \begin{cases}7, & if j = 0; \\ (3j + 7)/5, & if j ≡ 1 (mod 5); \\ (3j + 3)/5, & if j ≡ 4 (mod 5); \\ 12j/5 + 6, & if j ≡ 0 (mod 5), j > 0; \\ 1, & otherwise. \end{cases}

Further,

∑_{j=1}^{k}(−1)^{j−1}a_{j} = \begin{cases}−\frac{6k}{5}, & if k ≡ 0 (mod 10); \\ \frac{6k}{5} + 11, & if k ≡ 5 (mod 10); \\ −\frac{3(k−1)}{5} + 2, & if k ≡ 1 (mod 10); \\ \frac{3(k−1)}{5} + 9, & if k ≡ 6 (mod 10); \\ −\frac{3(k−2)}{5} + 1, & if k ≡ 2 (mod 10); \\ \frac{3(k−2)}{5} + 10, & if k ≡ 7 (mod 10); \\ −\frac{3(k−3)}{5} + 2, & if k ≡ 3 (mod 10); \\ \frac{3(k−3)}{5} + 9, & if k ≡ 8 (mod 10); \\ −\frac{6(k−4)}{5} − 1, & if k ≡ 4 (mod 10); \\ \frac{6(k−4)}{5} + 12, & if k ≡ 9 (mod 10). \end{cases}  (6)

In the same way as in the proof of Theorem 1 we observe |e^{2} − 7 − s′_{k}/t′_{k}| = O(1/k) and

\frac{t′_{k−1}}{t′_{k}} = O (\frac{1}{k}) + \begin{cases}0, & if k ≡ 0, 1, 4 (mod 5); \\ 1, & if k ≡ 2 (mod 5); \\ \frac{1}{2}, & if k ≡ 3 (mod 5). \end{cases}

Thereby, and by (6), we obtain the theorem.

Remark 4. 1. It is easy to see that the error term O(1/k) in the theorems cannot be made smaller. Accordingly, the convergence is rather slow, which is a further difference between the present cases and the case of quadratic irrationals.

2. The continued fraction expansions of e^{2/q} and (e^{2/q} + 1)/(e^{2/q} − 1) for integers q ≥ 1 have a shape similar to that of e, e^{2}, and e^{∗} [8, p. 124 f.]. The same holds for the the numbers tan(1/q). Therefore, similar theorems about Dedekind sums can be expected for the convergents of these numbers.

3. Due to a theorem of Hurwitz [8, p. 119] one may even hope for similar results for the numbers

\frac{ae^{2/q} + b}{ce^{2/q} + d},

where the integer q is ≥ 1 and a, b, c, d ∈ ℤ are such that ad − bc ≠ 0. It seems, however, that not all continued fraction expansions of these numbers are explicitly known.

4. The continued fraction expansions of the numbers

∑_{j=0}^{∞} b^{−2^{j}}, b ∈ ℤ, b ≥ 3,

are also known [10]. They are, however, much more involved than those considered here. Accordingly, the asymptotic behaviour of the corresponding Dedekind sums seems to be far more complicated.

References

[1] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer, 1976.

[2] Ph. Barkan, Sur les sommes de Dedekind et les fractions continues finies, C. R. Acad. Sci. Paris Sér. A-B 284 (1977) A923–A926.

[3] K. Girstmair, Dedekind sums in the vicinity of quadratic irrationals, J. Number Th. 132 (2012), 1788–1792.

[4] K. Girstmair and J. Schoißengeier, On the arithmetic mean of Dedekind sums, Acta Arith. 116 (2005), 189–198.

[5] D. Hickerson, Continued fractions and density results for Dedekind sums, J. Reine Angew. Math. 290 (1977), 113–116.

[6] D. E. Knuth, Notes on generalized Dedekind sums, Acta Arith. 33 (1977), 297–325.

[7] C. Meyer, Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, 1957.

[8] O. Perron, Die Lehre von den Kettenbrüchen, vol. I (3rd ed.), Teubner, 1954.

[9] H. Rademacher and E. Grosswald, Dedekind Sums, Mathematical Association of America, 1972.

[10] J. Shallit, Simple continued fractions for some irrational numbers, J. Number Th. 11 (1979), 209–217.

[11] J. Shallit, Origins of the analysis of the Euclidean algorithm, Hist. Math 21 (1994), 401–419.

[12] G. Urzúa, Arrangements of curves and algebraic surfaces, J. Algebraic Geom. 19 (2010), 335–365.

2010 Mathematics Subject Classification: Primary 11F20; Secondary 11A55. Keywords: asymptotic behaviour of Dedekind sums, continued fraction expansions of transcendental numbers.

Received May 8 2012; revised version received May 31 2012. Published in Journal of Integer Sequences, June 12 2012.

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