On some oscillating sums

Abstract

This paper deals with the sums S α (n)=∑ j=1 n (-1) ⌊jα⌋ where α is any real number. The interest in these sums was initiated by a problem proposed by H. D. Ruderman [Problem 6105, Am. Math. Mon. 83, 573 (1977)] and solved (among other) by D. Borwein [Solution to problem no. 6105. Am. Math. Mon. 85, 207–208 (1978)] that the series ∑ n=1 ∞ (-1) ⌊n2⌋ /n converges. It quickly turned out that the behavior of such sums is intimately connected with the simple continued fraction expansions α=[a 0 ;a 1 ,a 2 ,⋯] and β=α/2=[b 0 ;b 1 ,b 2 ,⋯]. E.g., P. Bundschuh [Arch. Math. 29, 518–523 (1977; Zbl 0365.10025)] proved that the series ∑ n=1 ∞ (-1) ⌊nα⌋ /n converges for numbers α with bounded b i of β=α/2=[b 0 ;b 1 ,b 2 ,⋯]. J. Schoissengeier [Unif. Distrib. Theory 2, 107–113 (2007; Zbl 1153.11033)] proved that the series ∑ n=1 ∞ (-1) ⌊nα⌋ /n and ∑ k=0,2∤q k ∞ (-1) k (logb k+1 )/q k converge simultaneously. Here p k q k are convergents of β=α/2=[b 0 ;b 1 ,b 2 ,⋯]. A. E. Brouwer and J. van de Lune [Math. Centrum, Amsterdam, Afd. zuivere Wisk. ZW 90/76, 16 p. (1976; Zbl 0359.10029)] have shown that S α (n)≥0 for all n if and only if the partial quotients a 2i of α=[a 0 ;a 1 ,a 2 ,⋯] are even for all i≥0. In this paper the authors study the sequence of those n for which S α (n) assumes a value for the first time, i.e., is larger/smaller than ever before, and it is denoted by t 0 =0,t 1 ,t 2 ,⋯. They show that for any irrational α the sum S α (n) is not bounded, so that the corresponding sequence t k actually is an infinite sequence. In a main result the authors prove that for every j≥1 there is an index k such that t j -t j-1 =Q k , where P k /Q k is a certain convergent of α=[a 0 ;a 1 ,a 2 ,⋯]. They also give explicit formulas for the numbers t j and the corresponding Q k . For quadratic irrational α the authors translate this into an algorithm which computes the sequence t n and S α (t n ) outputs [a 0 ;a 1 ,a 2 ,⋯]. In the final Part the authors summarize their open problems, including that t k is recurrent and the sequence sign(S(t k )) is to be purely periodic. In the Appendix the authors extend a fast algorithm in R. Fokkink, W. Fokkink and J. van de Lune [Nieuw Arch. Wiskd., IV. Ser. 12, 13–18 (1994; Zbl 0826.11061)] for the computation of S α (n) for any irrational α and for very large n in terms of β=α/2=[b 0 ;b 1 ,b 2 ,⋯]. E.g., S 2 (10 1000 )=-10, S 2 (10 10000 )=166, S π (10 10000 )=11726