Abstract
Using Hilbert transforms, we establish two families of sum rules involving Bessel moments, which are integrals associated with Feynman diagrams in two-dimensional quantum field theory. With these linear relations among Bessel moments, we verify and generalize two conjectures by Bailey–Borwein–Broadhurst–Glasser and Broadhurst–Mellit.
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Notes
For our purposes, it suffices to evaluate the Hilbert transform of a function for almost every (a.e.) real variable, leaving out a subset of measure zero that will not affect subsequent computations of Lebesgue integrals. Later afterwards, an equal sign may also be used to denote an equality that is valid almost everywhere (even when “a.e.” is not written), depending on context.
Confusingly, in [1, footnote 13], the authors apparently declared that the conjecture on \( Z_{2n,n-2k}=0\) “has also now been resolved,” without supplying further citations to publications or preprints. This contradicts the recent claims in [5, (112)] and [9, (5.6)] on the open status of \( Z_{6,1}=0.\)
Naïvely, upon observing that \( (n!)^2/2^n\in \mathbb Z,\,{\forall } n\in \mathbb Z_{\ge 4}\) and \( D_n\in \mathbb Z,\,{\forall } n\in \mathbb Z_{\ge 0},\) we obtain \( 2^{\ell -1}\alpha _\ell \in \mathbb Z,\,{\forall } \ell \in \mathbb Z_{>0},\) at best. The divisibility statement \( 4^{n}\mid (n!)^2D_n,\,{\forall } n\in \mathbb Z_{\ge 0}\) is thus deeper than these naïve observations. In our proof of the integrality \( \alpha _\ell \in \mathbb Z,\,{\forall } \ell \in \mathbb Z_{>0},\) we need Rogers’ work on modular forms [15], which in turn, was inspired by Bertin’s studies of modular parametrizations for certain families of Calabi–Yau manifolds [2].
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Acknowledgements
In early 2017, I wrote up this paper in Beijing, mostly drawing on my research notes prepared at Princeton during 2012 and 2013. I thank Prof. Weinan E for arranging my stays in Princeton and Beijing, as well as organizing a seminar on constructive quantum field theory at Princeton. After completion of the initial draft of this article, I received from Dr. David Broadhurst his slides for recent talks [6,7,8] on Bessel moments, which set his conjectures in a wider context. I thank Dr. Broadhurst for his constant encouragements and incisive comments on this project. I am indebted to an anonymous referee for thoughtful suggestions on improving the presentation of this paper. In January 2013, I benefited from fruitful discussions with Prof. Jon Borwein on his previous contributions to Bessel moments and elliptic integrals; I was equally grateful to his friendly communications on my then-unpublished work related to Hilbert transforms. I dedicate this work to his memory.
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To the memory of Jonathan M. Borwein (1951–2016)
This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).
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Zhou, Y. Hilbert transforms and sum rules of Bessel moments. Ramanujan J 48, 159–172 (2019). https://doi.org/10.1007/s11139-017-9945-y
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DOI: https://doi.org/10.1007/s11139-017-9945-y