Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

Keating, Jon P; Rodgers, Brad; Roditty-Gershon, Edva; Rudnick, Zeév

doi:10.1007/s00209-017-1884-1

Cited by 48 publications

(101 citation statements)

References 36 publications

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“…The other cases exhibit similar behavior, so we infer that the approximation P(c, α, β, n) is accurate when n > 3. The expression P(c, 0, 0, n) here gives an easy way to characterise the coefficients of P(c, 0, 0, n) conjectured in [24].…”

Section: )mentioning

confidence: 99%

Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions

Zhan¹,

Blower

Chen³

et al. 2018

Journal of Mathematical Physics

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In this paper, we study the probability density function, P(c, α, β, n) dc, of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trM n ∈ (c, c + dc), where M n are n × n matrices drawn from the unitary Jacobi ensembles. We first compute the exponential moment generating function of the linear statistics n j=1 f (x j ) := n j=1 x j , denoted by M f (λ, α, β, n). The weight function associated with the Jacobi unitary ensembles reads x α (1 − x) β , x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant D n (λ, α, β) generated by the time-evolved Jacobi weight, namely, w(x; λ, α,We think of λ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, P n (x, λ) = x n + p(n, λ) x n−1 + • • • + P n (0, λ), orthogonal with respect to w(x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor, Chen and Ehrhardt [5], with some change of variables, we obtain a certain auxiliary variable r n (λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w(x; λ, α, β)/x. It is shown that r n (2ie z ) satisfies a Chazy II equation. There is another auxiliary variable, denote as R n (λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w(x; λ, α, β)/x. Then Y n (−λ) = 1 − λ/R n (λ) satisfies a particular Painlevé V:The σ n function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo-Miwa-Okamoto σ−form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an

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Section: )mentioning

confidence: 99%

Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions

Zhan¹,

Blower

Chen³

et al. 2018

Journal of Mathematical Physics

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“…In the function field setting, the analogous problem was studied by Rodgers, Rudnick and the second and third authors [KRRGR18]. Let M n ⊂ F q [t] be the set of monic polynomials of degree n with coefficients in F q .…”

Section: Introductionmentioning

confidence: 99%

“…where I k (n; deg Q − 1) is a certain matrix integral over the unitary group U deg Q−1 (C) -see [KRRGR18] and (1.1.1) below. This integral can be evaluated in a number of ways.…”

Section: Introductionmentioning

confidence: 99%

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Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t]

Roditty-Gershon

Hall²,

Keating

2019

Int. J. Number Theory

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We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in Fq[t], in the limit as q → ∞. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when q → ∞, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual k-divisor function, when the L-function in question has degree one. They illustrate the role played by the degree of the L-functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over Fq[t], and we illustrate them by examining in some detail the generalised k-divisor functions associated with the Legendre curve.

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“…The function field analog of our setup, where the digits are replaced by the coefficients of a polynomial (1.3) m(T ) = a 0 + a 1 T + · · · + a n−1 T n−1 + a n T n over a finite field F q , has also received a lot of attention, as can be seen from [1,3,4,10,11,14,15,21,23,24,25,26,29,30,31,32,33,38,39,41,42,43,44,45,46,47,48,53,54,55,57]. In this work we contribute to the study of the function field analog by giving lower bounds on the number of squarefrees in 'sparse' boxes.…”

Section: Introductionmentioning

confidence: 99%

Squarefree polynomials with prescribed coefficients

Oppenheim

Shusterman

2018

Journal of Number Theory

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Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

Cited by 48 publications

References 36 publications

Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions

Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions

Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t]

Squarefree polynomials with prescribed coefficients

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