Dedekind Sums with Arguments Near Euler s Number e

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1 Journal of Integer Sequences, Vol. 1 (01), Article 1..8 Dedeind Sums with Arguments Near Euler s Number e Kurt Girstmair Institut für Mathemati Universität Innsbruc Technierstr. 1/7 A-600 Innsbruc Austria Kurt.Girstmair@uib.ac.at Abstract We study the asymptotic behaviour of the classical Dedeind sums s(m/n) for convergents m/n of e, e, and (e+1)/(e 1), where e = is Euler s number. Our main tool is the Baran-Hicerson-Knuth formula, which yields a precise description of what happens in all cases. 1 Introduction and results Dedeind 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, 1]). Let n be a positive integer and m Z, (m,n) = 1. The classical Dedeind sum s(m/n) is defined by n s(m/n) = ((/n))((m/n)) =1 where ((...)) is the usual sawtooth function (for example, [9, p. 1]). In the present setting it is more natural to wor with instead. S(m/n) = 1s(m/n) 1

2 In the previous paper [] we used the Baran-Hicerson-Knuth-formula to study the asymptotic behaviour of S(s /t ) for the convergents s /t of a periodic simple continued fraction α = [a 0,a 1,a,...], i. e., for a quadratic irrational α. In this situation two cases are possible: The sequence S(s /t ) either remains bounded with a finite number of cluster points or it essentially behaves lie C for some constant C depending on α. In the latter case S(s /t ) C 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 n on average [4], quadratic irrationalities produce Dedeind sums of a considerably smaller size. In fact, the inequality log t / log + 1 was already proved in 1841 [11]. Accordingly, if S(s /t ) is not bounded, we have S(s /t ) = O(log t ) for a quadratic irrational α. Because the structure of the continued fraction expansions of transcendental numbers lie e or e is similar to that of quadratic irrationals [8, p. 1 ff.], nothing prevents us from applying the Baran-Hicerson-Knuth-formula (() below) to these cases. It turns out that the asymptotic behaviour of Dedeind sums is quite similar to the case of quadratic irrationals. Only the case S(s /t ) 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 put, if 0, 1, (mod 6); L() =, otherwise. Then we have, for the convergents s /t of Euler s number e, S(s /t ) L() = O e, if (mod 6); e, if 1 (mod 6); 6 + e +, if (mod 6); e +, if 4 (mod 6); 6 e, if (mod 6). The continued fraction expansion of e = is more complicated than that of e. This has the effect that the analogue of Theorem 1 also loos more complicated. We obtain Theorem. For a nonnegative integer put, if 1,, (mod 10);, if 6, 7, 8 (mod 10); L() = 6, if 0, 4 (mod 10); 6, if, 9 (mod 10).

3 Then we have, for the convergents s /t of the number e, e 7, if 0 (mod 10); S(s /t ) L() = O + e 7, if 1 (mod 10); e 9, if (mod 10); e 1 + 1, if (mod 10); e 16, if 4 (mod 10); e + 1, if (mod 10); e + 7, if 6 (mod 10); e 1, if 7 (mod 10); e 4 + 1, if 8 (mod 10); e 14, if 9 (mod 10). Finally, we consider the case of e = (e + 1)/(e 1), which is fairly simple. Theorem. For a nonnegative integer put, if is even; L() =, if is odd. Then we have, for the convergents s /t of e, Proofs S(s /t ) L() = O + e, if is even; e 1, if is odd. We start with the continued fraction expansion [a 0,a 1,a,...] of an arbitrary irrational number. The numerators and denominators of its convergents s /t are defined by the recursion formulas s = 0, s 1 = 1, s = a s 1 + s and t = 1, t 1 = 0, t = a t 1 + t, for 0. (1) The Baran-Hicerson-Knuth formula says that for 0 (s S(s /t ) = ( 1) j 1 + t 1 )/t, if is odd; a j + () (s t 1 )/t, if is even; [], [], [6]. Here s and t are defined as in (1), but for the number [0,a 1,a,...] instead of [a 0,a 1,a,...]. We prove the simplest case first.

4 Proof of Theorem. The digits a j of the continued fraction expansion of e are a j = 4j +, j = 0, 1,,... [8, p. 14]. An easy calculation shows that for 0 ( 1) j 1, if is even; a j = () + 4, if is odd. Now s /t converges against [0,a 1,a,...] = e, and e s /t < 1/t [8, p. 7]. We remared in the Introduction that = O(log t ). Hence we also have e s /t = O(1/). Finally, (1) gives t 1 /t = t 1 /(a t 1 + t ) 1/a = O(1/). These observations, together with () and (), prove the theorem. Proof of Theorem 1. In the case of e = [a 0,a 1,a,...] one easily derives from [8, p. 14] that, if j = 0; a j = (j 1)/ +, if j (mod ); 1, otherwise. An elementary computation with arithmetic series (which is more laborious than that of the proof of Theorem ) yields, if 0 (mod 6); ( 1) j 1 a j = 1 + 1, if (mod 6); + 1, if 1 (mod 6);, if 4 (mod 6); 1 1, if (mod 6); +, if (mod 6). In the same way as in the proof Theorem we have s /t e and e s /t = O(1/). If (mod ), we note t 1 /t 1/a = O(1/). If 0 (mod ) and, we have (4) t 1 t = t 1 t 1 t = 1. () 1 + t /t 1 Since t /t 1 = O(1/), this shows t 1 /t = 1 + O(1/). If 1 (mod ) and 4, formula () also holds. Together with t /t 1 = 1+O(1/), it gives t 1 /t = 1/+O(1/). These observations, combined with () and (4), prove the theorem. Proof of Theorem. The proof follows the above pattern. One obtains from [8, p. 1] 7, if j = 0; (j + 7)/, if j 1 (mod ); a j = (j + )/, if j 4 (mod ); 1j/ + 6, if j 0 (mod ),j > 0; 1, otherwise. 4

5 Further, ( 1) +, if 1 (mod 10); ( 1) + 9, if 6 (mod 10); ( ) + 1, if (mod 10); ( 1) j 1 a j = ( ) + 10, if 7 (mod 10); ( ) +, if (mod 10); ( ) + 9, if 8 (mod 10); 6 6, if 0 (mod 10); + 11, if (mod 10); 6( 4) 1, if 4 (mod 10); 6( 4) + 1, if 9 (mod 10). In the same way as in the proof of Theorem 1 we observe e 7 s /t = O(1/) and t 1 t = O + Thereby, and by (6), we obtain the theorem. 0, if 0, 1, 4 (mod ); 1, if (mod );, if (mod ). 1 Remar It is easy to see that the error term O(1/) 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.. The continued fraction expansions of e /q and (e /q + 1)/(e /q 1) for integers q 1 have a shape similar to that of e, e, and e [8, p. 14 f.]. The same holds for the the numbers tan(1/q). Therefore, similar theorems about Dedeind sums can be expected for the convergents of these numbers.. Due to a theorem of Hurwitz [8, p. 119] one may even hope for similar results for the numbers ae /q + b ce /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 nown. 4. The continued fraction expansions of the numbers b j, b Z,b, j=0 are also nown [10]. They are, however, much more involved than those considered here. Accordingly, the asymptotic behaviour of the corresponding Dedeind sums seems to be far more complicated. (6)

6 References [1] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer, [] Ph. Baran, Sur les sommes de Dedeind et les fractions continues finies, C. R. Acad. Sci. Paris Sér. A-B 84 (1977) A9 A96. [] K. Girstmair, Dedeind sums in the vicinity of quadratic irrationals, J. Number Th. 1 (01), [4] K. Girstmair and J. Schoißengeier, On the arithmetic mean of Dedeind sums, Acta Arith. 116 (00), [] D. Hicerson, Continued fractions and density results for Dedeind sums, J. Reine Angew. Math. 90 (1977), [6] D. E. Knuth, Notes on generalized Dedeind sums, Acta Arith. (1977), 97. [7] C. Meyer, Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlörpern, Aademie-Verlag, 197. [8] O. Perron, Die Lehre von den Kettenbrüchen, vol. I (rd ed.), Teubner, 194. [9] H. Rademacher and E. Grosswald, Dedeind Sums, Mathematical Association of America, 197. [10] J. Shallit, Simple continued fractions for some irrational numbers, J. Number Th. 11 (1979), [11] J. Shallit, Origins of the analysis of the Euclidean algorithm, Hist. Math 1 (1994), [1] G. Urzúa, Arrangements of curves and algebraic surfaces, J. Algebraic Geom. 19 (010), Mathematics Subject Classification: Primary 11F0; Secondary 11A. Keywords: asymptotic behaviour of Dedeind sums, continued fraction expansions of transcendental numbers. Received May 8 01; revised version received May Published in Journal of Integer Sequences, June Return to Journal of Integer Sequences home page. 6