Period Functions for Maass Wave Forms. I

Lewis, James B.; Zagier, Don

doi:10.2307/2661374

Cited by 152 publications

(315 citation statements)

References 7 publications

(16 reference statements)

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“…Each of these transforms is characterised by regularity properties and a functional equation. Lastly and most importantly, let us point out that, surprisingly, there are striking similarities and analogies between the results proved here as well as in [2,3] with Lewis-Zagier's [28] results on period functions for Maass wave forms. That work is an expanded and clarified exposition of an earlier paper by Lewis [27].…”

Section: Motivation and Description Of Resultsmentioning

confidence: 89%

The Moments of Minkowski Question Mark Function: The Dyadic Period Function

Alkauskas

2009

Glasgow Math. J.

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Abstract. The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane ‫ރ‬ \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert-Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern-Brocot tree. Surprisingly, the Eisenstein series G 2 (z) does manifest in both real and p-adic cases.

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Section: Motivation and Description Of Resultsmentioning

confidence: 89%

The Moments of Minkowski Question Mark Function: The Dyadic Period Function

Alkauskas

2009

Glasgow Math. J.

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“…2. The period function P associated with a cusp form in S r ( , v) has properties analogous to those of the period function described by John Lewis and Don Zagier [9].…”

Section: Smaller Modulesmentioning

confidence: 81%

Holomorphic modular forms and cocycles

Bruggeman

2018

Res Math Sci

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“…Specifically, for λ = 1, (27) is known as the Lewis equation and has been studied (for complex β) because of its connection to the Selberg ζ-function and period polynomials (cusp forms of the modular group) [13]. An operator related to K β (22) also appears in this context and is called the Mayer operator [14].…”

Section: Transfer Operator Approachmentioning

confidence: 99%

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Fiala

Kleban

Özlük

2003

Journal of Statistical Physics

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We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (RuellePerron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form C ∼ [ǫ ln 2 ǫ] −1 . The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.

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Period Functions for Maass Wave Forms. I

Cited by 152 publications

References 7 publications

The Moments of Minkowski Question Mark Function: The Dyadic Period Function

The Moments of Minkowski Question Mark Function: The Dyadic Period Function

Holomorphic modular forms and cocycles

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