Some sums involving Farey fractions I

Kanemitsu, Shigeru; Rao, R. Sita Rama Chandra; Sarma, Abinash

doi:10.2969/jmsj/03410125

Cited by 13 publications

(16 citation statements)

References 9 publications

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“…Our object here is to investigate the distribution of spacings between Farey points in subintervals of [0,1]. Various results related to this problem have been obtained by [2,3,[5][6][7][8][10][11][12][13].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Since T k is a polygon which is independent of q and has at most 4 h−1 sides, the definition of J k,B (Q, q) from (5•10) shows that for each k ∈ K(B), there exists a partition of [1, Q] into N (k) intervals such that the function f k,B is monotonic on each of these intervals. Therefore, the variation of 11) for some constant C(B) > 0. We now combine (5•8) and (5•11) to get x = 1, both with y ∈ 2/(k + 1), 2/k , and respectively x + y = 1, (k + 1)y − x = 1, both with y ∈ 2/(k + 2), 2/(k + 1) ; see the picture in Figure 3.…”

Section: The Case Of the Full Interval [0 1]mentioning

confidence: 99%

“…Yet this is a very strong statement, as Franel and Landau [3,4] have shown that (1·3) is equivalent to the Riemann Hypothesis. Various results related to this problem have been obtained by [2,3,[5][6][7][8][10][11][12][13]. Various results related to this problem have been obtained by [2,3,[5][6][7][8][10][11][12][13].…”

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confidence: 98%

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The h-spacing distribution between Farey points

Augustin

Boca

Cobeli

et al. 2001

Math. Proc. Camb. Phil. Soc.

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Let I = [α, β] be a subinterval of [0, 1]. For each positive integer Q, we denote by [Fscr ]I(Q) the set of Farey fractions of order Q from I, that isand order increasingly its elements γj = aj/qj as α [les ] γ1 < γ2 < … < γNI(Q) [les ] β. The number of elements of [Fscr ]I(Q) isWe simply let [Fscr ](Q) = [Fscr ][0,1](Q), N(Q) = N[0,1](Q).Farey sequences have been studied for a long time, mainly because of their role in problems related to diophantine approximation. There is also a connection with the Riemann zeta function which has motivated their study. Farey sequences seem to be distributed as uniformly as possible along [0, 1]; a way to prove it is to show thatfor all ε > 0, as Q → ∞. Yet this is a very strong statement, as Franel and Landau [3, 4] have shown that (1·3) is equivalent to the Riemann Hypothesis.Our object here is to investigate the distribution of spacings between Farey points in subintervals of [0, 1]. Various results related to this problem have been obtained by [2, 3, 5–8, 10–13].

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: The Case Of the Full Interval [0 1]mentioning

confidence: 99%

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The h-spacing distribution between Farey points

Augustin

Boca

Cobeli

et al. 2001

Math. Proc. Camb. Phil. Soc.

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“…We may apply the inclusion-exclusion principle similar to that in the proof of Lemma 5 [5] to deduce the recurrence…”

Section: Proposition 1 With A(q) As Defined By (238) We Havementioning

confidence: 99%

On the discrete mean square of Dirichlet L-functions at 1

et al. 2004

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There have been many results obtained so far for the mean square of the (absolute) value of the Dirichlet L-function L(s, χ ) in the critical strip 0 < σ < 1, especially on the critical line Re s = 1 2 , but relatively few results were known for discrete mean value of |L(1, χ)| 2 till W. Zhang had published papers improving the error term step by step, which have recently been superseded by M. Katsurada and K.Matsumoto in which they succeeded in deriving an "asymptotic formula" for χ =χ 0 |L(1, χ)| 2 .The object of our paper is to point out a structural property contained in the formation of the mean square, to find out the niryana-the true body of the above sum.

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“…In a number of papers including Hall (1970) and Kanemitsu et al (1982), the limiting behaviour of σ (n) β is studied in the case when {x i,n } is the Farey sequence of order n, that is, the set of all fractions p/q in [0, 1] with gcd(p, q) = 1 and q ≤ n. In the present paper we study the limiting behaviour of σ (n) β when x i,n are the elements of F n ; in this case we will write σ…”

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confidence: 99%