The Mahler measure of a Calabi–Yau threefold and special $$L$$ L -values

Papanikolas, Matthew A.; Rogers, Mathew D.; Samart, Detchat

doi:10.1007/s00209-013-1238-6

Cited by 5 publications

(3 citation statements)

References 29 publications

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“…Given for any weight 4 modular form f, the functional equation relates L(f, s) and L(f, 4 − s), the value at s = 4 may have special importance. Papanikolas, Rogers and Samart (2014) and Wan and Zucker (2016) have derived hypergeometric series expressions for the L-series of f 4, 8 evaluated at 4, one being…”

Section: Discussionmentioning

confidence: 99%

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

McCrorie

2020

Sankhya B

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This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants representable as contiguous very well-poised generalized hypergeometric series and as even moments of the square of the complete elliptic integral of the first kind. Neither constants are currently available in closed form. New symmetries are found in the critical values of the L-series of two underlying cusp forms, providing a sense in which one of the constants has a formal counterpart. The significant roles this constant and its counterpart play in multidisciplinary contexts is described. The results unblock the problem of representing them in terms of lower-order generalized hypergeometric series, offering progress towards identifying their closed forms. The same approach facilitates a canonical characterization of the hypergeometry of the parbelos, adding to the characterizations outlined by Campbell, D'Aurozio and Sondow (2020, The American Mathematical Monthly127(1), 23-32). The paper also connects the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis to very well-poised generalized hypergeometric series. The confluence of ideas presented reflects a multidisciplinarity that accords with the approach and philosophy of Prasanta Chandra Mahalanobis.

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Section: Discussionmentioning

confidence: 99%

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

McCrorie

2020

Sankhya B

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“…Other deeper such formulae involve L-functions of various kinds, evaluated at specific integers. See the survey of Bertin and Lalín [1], and also Papanikolas et al [16] for a more recent example. There may even be some connection between Mahler measure of integer polynomials and the Feynman integrals of mathematical physics -see for instance Samart [17] and Vanhove [23].…”

Section: Introductionmentioning

confidence: 99%

Closed sets of Mahler measures

Smyth

2018

Proc. Amer. Math. Soc.

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Abstract. Given a k-variable Laurent polynomial F , any ℓ×k integer matrix A naturally defines an ℓ-variable Laurent polynomial F A . I prove that for fixed F the set M(F ) of all the logarithmic Mahler measures m(F A ) of F A for all A is a closed subset of the real line. Moreover, the matrices A can be assumed to be of a special form, which I call Saturated Hermite Normal Form. Furthermore, if F has integer coefficients and M(F ) contains 0, then 0 is an isolated point of this set.I also show that, for a given bound B > 0, the set M B of all Mahler measures of integer polynomials in any number of variables and having length (sum of the moduli of its coefficients) at most B is closed. Again, 0 is an isolated point of M B .These results constitute evidence consistent with a conjecture of Boyd from 1980 to the effect that the union L of all sets M B for B > 0 is closed, with 0 an isolated point of L.

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“…have appeared in the context of multi-loop Feynman diagrams, Ising-type integrals [2], random walks [5], Mahler measures, and non-critical L-values of modular forms [8,9]. More intricate integrals containing K also appear in connection with lattice Green's functions [4,14].…”

Section: Introductionmentioning

confidence: 99%

Integrals of K and E from lattice sums

Wan

Zucker

2015

Ramanujan J

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We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.

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The Mahler measure of a Calabi–Yau threefold and special $$L$$ L -values

Cited by 5 publications

References 29 publications

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

Closed sets of Mahler measures

Integrals of K and E from lattice sums

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