Asymptotics of class numbers for progressions and for fundamental discriminants

Raulf, Nicole

doi:10.1515/forum.2009.012

Cited by 11 publications

(14 citation statements)

References 5 publications

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“…The proof of the last statement is analogous to the proof of the first statement when we use (20) and Lemma 2.…”

Section: éTienne Fouvry and Jürgen Klünersmentioning

confidence: 93%

“…One can summarize Sarnak's result in saying that H(d) and d have the same order of magnitude, when one uses the ordering by the size of d . (For a generalization and a transposition of the results of [21] to the case of arithmetic progressions and of fundamental discriminants, see [20]). In [22, Conjecture 1], Sarnak also proposes a conjecture to guess the statistical size of h(D) with D fundamental.…”

Section: Applications To the Class Number Of Real Quadratic Fieldsmentioning

confidence: 99%

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Weighted Distribution of the 4-rank of Class Groups and Applications

Fouvry¹,

Klüners²

2010

International Mathematics Research Notices

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Abstract. We prove that the distribution of the values of the 4-rank of ideal class groups of quadratic fields is not affected when it is weighted by a divisor type function. We then give several applications concerning a new lower bound of the sums of class numbers of real quadratic fields with discriminant less than a bound tending to infinity and several questions of P. Sarnak concerning reciprocal geodesics.

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“…The proof of the last statement is analogous to the proof of the first statement when we use (20) and Lemma 2.…”

Section: éTienne Fouvry and Jürgen Klünersmentioning

confidence: 93%

Section: Applications To the Class Number Of Real Quadratic Fieldsmentioning

confidence: 99%

Weighted Distribution of the 4-rank of Class Groups and Applications

Fouvry¹,

Klüners²

2010

International Mathematics Research Notices

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“…One can obtain similar results for the corresponding moments of class numbers over fundamental discriminants, by using the techniques of this paper together with [11]. Note that when d is fundamental, h(d) is the narrow class number of the real quadratic field Q( √ d), which equals the class number of Q( √ d) if the negative Pell equation t 2 − du 2 = −4 is solvable, and equals twice this class number if the negative Pell equation is not solvable.…”

Section: Introductionmentioning

confidence: 57%

“…We shall prove that summing over d ∈ D such that ε d ≤ x amounts to essentially summing over the corresponding pairs (t, u) in a certain range. This follows from the arguments in [13] and was used by Raulf [11] to compute the first moment of class numbers over fundamental discriminants. Here and throughout we define…”

mentioning

confidence: 99%

Large Moments and Extreme Values of Class Numbers of Indefinite Binary Quadratic Forms

Lamzouri

2017

Mathematika

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Let h(d) be the class number of indefinite binary quadratic forms of discriminant d, and let ε d be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the k-th moment of h(d) over positive discriminants d with ε d ≤ x, uniformly for real numbers k in the range 0 < k ≤ (log x) 1−o(1) . This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer k. We also investigate the distribution of large values of h(d) when the d's are ordered according to the size of their fundamental units ε d . In particular, we show that the tail of this distribution has the same shape as that of class numbers of imaginary quadratic fields ordered by the size of their discriminants. As an application of these results, we prove that there are many positive discriminants d with class number h(d) ≥ (e γ /3 + o(1)) · ε d (log log ε d )/ log ε d , a bound that we believe is best possible. We also obtain an upper bound for h(d) that is twice as large, assuming the generalized Riemann hypothesis.

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“…According to Lemma 2.11 and 2.19 in [16], we see that The following lemma describes the values in Lemma 3.5 for ΓΓ(p r ) = SL 2 (Z).…”

Section: )mentioning

confidence: 90%

Correlations of multiplicities in length spectra for congruence subgroups

Hashimoto

2012

Bulletin of the London Mathematical Society

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Bogomolny- Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup Γ 0 (n) and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group.

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Asymptotics of class numbers for progressions and for fundamental discriminants

Cited by 11 publications

References 5 publications

Weighted Distribution of the 4-rank of Class Groups and Applications

Weighted Distribution of the 4-rank of Class Groups and Applications

Large Moments and Extreme Values of Class Numbers of Indefinite Binary Quadratic Forms

Correlations of multiplicities in length spectra for congruence subgroups

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