On the Length of the Period of a Quadratic Irrationality

Golubeva, E. P.

doi:10.1070/sm1985v051n01abeh002850

Cited by 7 publications

(8 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [8] (see also [6]) E.P. Golubeva proves that the left-hand side of this inequality is asymptotically small in comparison with the right-hand one; moreover, if the extended Riemann conjecture is true, then the order of their difference is not less than ln(Q) ln 2−ε .…”

Section: Statement and Discussion Of Obtained Resultsmentioning

confidence: 95%

About statistics of periods of continued fractions of quadratic irrationalities

Lerner

2010

Funct. Anal. Other Math.

View full text Add to dashboard Cite

In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form x 2 + px = q with integer p and q, p 2 + q 2 ≤ R 2 . We prove a weak variant of Arnold conjectures about the Gauss-Kuzmin statistics with R → ∞. Statement and discussion of obtained resultsAny value x ∈ R is representable as a continued fraction (CF)where a 0 ∈ Z (if a 0 = 0, then we omit it), a i ∈ N for all i ≥ 1. Let i be a fixed position in the CF expansion and let P i (k) stand for the probability that a i = k with x randomly chosen in the segment [0, 1). Hereinafter we understand the random choice as a realization of the uniform distribution. It is well known that according to the Gauss-Kuzmin theorem,The roots of the quadratic equation x 2 + px = q, where p, q ∈ Z, = p 2 + 4q > 0, are expandable in a periodic CF. Let x + (p, q) = √ /2 − p/2. The fractional part of this value

show abstract

Section: Statement and Discussion Of Obtained Resultsmentioning

confidence: 95%

About statistics of periods of continued fractions of quadratic irrationalities

Lerner

2010

Funct. Anal. Other Math.

View full text Add to dashboard Cite

show abstract

“…As already mentioned above, the length s = ~(d) of the period and the corresponding unit e(d) are connected by the estimates (see [4], where an upper estimate has been obtained, and [5], where the second part of the inequality has been proved).…”

Section: Tabular Data For the Lengths Of Periodsmentioning

confidence: 90%

“…We note that on the average with respect to @ (see [5]) we have 6<~)~, more precisely, so that discriminants for which ~(~}~-~ ~ , are rare. In the paper one computes the Fourier coefficients of the Eisenstein series of the orthogonal group of signature (i, 4).…”

Section: Tabular Data For the Lengths Of Periodsmentioning

confidence: 96%

“…It is known [4], [5] that, roughly speaking, log e(d) coincides with the length s of the period of this expansion (more precisely, the difference between them is negligibly small in comparison with the abyss which separates the existing results from the real state of affairs; see Sec. It is known [4], [5] that, roughly speaking, log e(d) coincides with the length s of the period of this expansion (more precisely, the difference between them is negligibly small in comparison with the abyss which separates the existing results from the real state of affairs; see Sec.…”

mentioning

confidence: 91%

“…However, by virtue of the results of [5], the sequences of the partial fractions of the expansion of /d into a continued fraction, where d is sufficiently large, while h(d) is small, have an extremely irregular structure. However, by virtue of the results of [5], the sequences of the partial fractions of the expansion of /d into a continued fraction, where d is sufficiently large, while h(d) is small, have an extremely irregular structure.…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Golubeva¹

1990

J Math Sci

View full text Add to dashboard Cite

The fundamental result of the paper is the following theorem: suppose that the Riemann conjecture is valid for the Dedekind ~-functions of all fields G( __i_ _(I+s ~ ~/k)-Then there exists a constant C > 0 such that on the interval p ~ x one can find at least Cx log -I x prime numbers p for which h(Sp 2) = 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 ~ ~(Sp 2) s p = 0 Cx ~/2) p4x 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.The Gauss--Hasse conjecture on the existence of an infinite number of real quadratic fields of the same class is a striking example of a problem in which number theory has made very little progress since Gauss' time. Lately a number of works have been published in which this conjecture is formulated in a more precise (containing quantitative statements) form (see [i], [2]), but, just as before, something definite can be said about the number h(d) of classes of binary quadratic forms of discriminant d (d > 0, d is not a perfect square) only in those cases when the fundamental unit e(d) of the corresponding order is explicitly computed. An exception is the paper [3], where the considered problem is formulated in terms which go beyond the framework of the present investigation. The unique tool for the computation of ~(d) is the expansion of ~d in a continued fraction. It is known [4], [5] that, roughly speaking, log e(d) coincides with the length s of the period of this expansion (more precisely, the difference between them is negligibly small in comparison with the abyss which separates the existing results from the real state of affairs; see Sec. 3). The quantity s behaves itself in an extremely irregular manner (see Table 1 in Sec. 3). In this sense it reminds us of the length of the period of the expansion of a rational number into a decimal fraction. We note that this analogy is not terminological since both problems are connected with the problemof primitive roots. In the case of decimal fractions the answer to the problem (belonging to Gauss) is given by Artin's conjecture, while in the case of a continued fraction for discriminants of a special form (for example, d = 5p =, p is prime) by its analogue for the corresponding quadratic field (see Sec. 2, where Hooley's results are carried over to the case of the field ~(~) ).

show abstract

Untitled

Golubeva¹

2001

Journal of Mathematical Sciences

View full text Add to dashboard Cite

On the Length of the Period of a Quadratic Irrationality

Cited by 7 publications

References 2 publications

About statistics of periods of continued fractions of quadratic irrationalities

About statistics of periods of continued fractions of quadratic irrationalities

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

Untitled

Contact Info

Product

Resources

About