On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction.

Let the continued fraction expansion of any irrational number be denoted by and let the -th convergent of this continued fraction expansion be denoted by . Let

where . Let . It is shown that if , then the Rogers-Ramanujan continued fraction diverges at . is an uncountable set of measure zero. It is also shown that there is an uncountable set of points such that if , then does not converge generally.

It is further shown that does not converge generally for 1$">. However we show that does converge generally if is a primitive -th root of unity, for some . Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

     

On the approximation by three consecutive continued fraction convergents

Denote by _pn/_qn,n=1,2,3,…$p_n/q_n,n=1,2,3,\dots$, the sequence of continued fraction convergents of the real irrational number x$x$. Define the sequence of approximation coefficients by _θn:=_qn|_qnx−_pn|,n=1,2,3,…${\theta }_n:=q_n|q_{n}x-p_n|,n=1,2,3,\dots$. A laborious way of determining the mean value of the sequence |_θn+1−_θn−1|,n=2,3,…$|{\theta }_{n+1}-{\theta }_{n-1}|,n=2,3,\dots$, is simplified. The method involved also serves for showing that for almost all x$x$ the pattern _θn−1<_θn<_θn+1${\theta }_{n-1}<{\theta }_n<{\theta }_{n+1}$ occurs with the same asymptotic frequency as the pattern _θn+1<_θn<_θn−1${\theta }_{n+1}<{\theta }_n<{\theta }_{n-1}$, namely 0.12109?$0.12109?$. All the four other patterns have the same asymptotic frequency 0.18945?$0.18945?$. The constants are explicitly given.     

On the Rogers-Ramanujan continued fraction

In the “Lost” note book, Ramanujan had stated a large number of results regarding evaluation of his continued fraction for certain values of τ. It is shown that all these results and many more have their source in the Kronecker limit formula.     

An algorithm of infinite sums representations and Tasoev continued fractions

For any given real number, its corresponding continued fraction is unique. However, given an arbitrary continued fraction, there has been no general way to identify its corresponding real number. In this paper we shall show a general algorithm from continued fractions to real numbers via infinite sums representations. Using this algorithm, we obtain some new Tasoev continued fractions.

     

An idea on some of Ramanujan’s continued fraction identities

We present an idea on how Ramanujan found some of his beautiful continued fraction identities. Or more to the point: why he chose the ones he wrote down among all possible identities.      

Metrical diophantine approximation for continued fraction like maps of the interval

We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

     

On sums of degrees of the partial quotients in continued fraction expansions of Laurent series

For any formal Laurent series with coefficients cn lying in some given finite field, let x=[a₀(x);a₁(x),a₂(x),…] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.     

Dynamics of a Ramanujan-type continued fraction with cyclic coefficients

We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters (Borwein et al., Exp. Math. 13, 275–286, 2004, Ramanujan J., in press, 2004; Borwein and Crandall, Exp. Math. 12, 287–296, 2004). A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in 2004, we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions. Russell Luke’s work was supported in part by a postdoctoral fellowship from the Pacific Institute for the Mathematical Sciences at Simon Fraser University.     

On the divergence of a certain series

We investigate for which real numbers α the series (4) converges, and prove that, even though it converges almost everywhere in the sense of Lebesgue to a periodic, with a period 1, odd function in L₂([0,1]), it is divergent at uncountably many points, the set of which is dense in [0,1]. Finally, we find the Fourier expansion of the function defined by the series (4).     

The Distribution of Reciprocal Pairs Modulo Polynomials over a Finite Field

Given a finite field F_q of order q, a fixed polynomial g in –F_q[X] of positive degree, and two elements u and v in the ring of polynomials in R = F_q [X]/gF_q[X], the question arises: How many pairs (a, 6) are there in R × R so that ab ? 1 mod g and so that a is close to u while b is close to v ? The answer is, about as many as one would expect. That is, there are no favored regions in R × R where inverse pairs cluster. The error term is quite sharp in most cases, being comparable to what would happen with random distribution of pairs. The proof uses Kloosterman sums and counting arguments. The exceptional cases involve fields of characteristic 2 and composite values of g. Even then the error term obtained is nontrivial. There is no computational evidence that inverses are in fact less evenly distributed in this case, however.     

A note on bounds for the derivatives of continued fractions

Although it is difficult to differentiate analytic functions defined by continued fractions, it is relatively easy in some cases to determine uniform bounds on such derivatives by perceiving the continued fraction as an infinite composition of linear fractional transformations and applying an infinite chain rule for differentiation.     

Convergence and divergence of the Ramanujan AGM fraction

We prove that the Ramanujan AGM fraction diverges if |a|=|b| with a ²≠b ². Thereby we prove two conjectures posed by J. Borwein and R. Crandall. We also demonstrate a method for accelerating the convergence of this continued fraction when it converges.      

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.     

Itinerary of a discontinuous map from the continued fraction expansion

A piecewise linear, discontinuous one-dimensional map is analyzed combinatorically. The quasi-periodic dynamics generated by iterations are completely characterized by successive convergents of a continued fraction associated with slopes of the map.     

Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

The fundamental result of the paper is the following theorem: suppose that the Riemann conjecture is valid for the Dedekind ζ-functions of all fields Then there exists a constant C>0 such that on the interval p≤x one can find at least Cx log⁻¹ x prime numbers p for which h(5p²)=2. Here h(d) is the number of proper equivalence classes of primitive binary quadratic forms of discriminant d. In addition, it is proved that . For these sequence of discriminants of a special form with increasing square-free part, one has obtained a nontrivial estimate from above for the number of classes. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 72–81, 1987.     

Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

It is proved that the relation h(d)=2 is valid for at least Cx^1/2 log^–2 x values of dx. Here h(d) is the number of the classes of binary quadratic forms of determinant d, while C>0 is a constant. Further, it is shown that for almost all primes p3 (mod 4), px, for (p), a fundamental unit of field and l(p), the length of the period of the continued fraction expansion of p, we have estimates (p)p² log^–c p, l(p)log p, which improve a result of Hooley (J. Reine Angew. Math., Vol. 353, pp. 98–131, 1984; MR 86d:11032). In addition, a generalization is given to composite discriminants of the Hirzebruch-Zagier formula, relating h(–p), p 3 (mod 4), with the continued fraction expansion of _p (Astérisque, no. 24–25, pp. 81–97, 1975; MR 51 #10293).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 168, pp. 11–22, 1988.     

Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences

The main theorem of this paper, proved using Mahler's method, gives a necessary and sufficient condition for the values Θ(x,a,q) at any distinct algebraic points to be algebraically independent, where Θ(x,a,q) is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives Θ(x,a,q) taking algebraically independent values for any distinct triplets (x,a,q) of nonzero algebraic numbers. Moreover, Θ(a,a,q) is expressed as an irregular continued fraction and Θ(x,1,q) is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively.     

On the fourth power mean of the generalized quadratic Gauss sums

The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q (a positive odd number), and give an exact computational formula for it.     

On the metric theory of the nearest integer continued fraction expansion

Supposek _n denotes either (n) or (p _n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp _k denotes thek ^th rationals prime. Also let denote the sequence of convergents to a real numberx and letc _n(x)) _n=1 be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants _n(x)) _n=1 by