Log-Sine Evaluations of Mahler Measures

Borwein, Jonathan M.; Straub, Armin

doi:10.1017/s1446788712000067

Cited by 17 publications

(33 citation statements)

References 17 publications

(58 reference statements)

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“…More generally, as shown in [12], immediately from (70) and the definition (6) of the log-sine integrals, we have…”

Section: Multiple Mahler Measuresmentioning

confidence: 85%

“…, 1 + x + y k ) (5) and provided an evaluation of µ 2 (1+x+y * ). As observed in [12] this Mahler measure can be conveniently expressed in terms of log-sine integrals. Namely, if…”

Section: Introductionmentioning

confidence: 94%

“…More examples of multiple Mahler measures that allow convenient evaluation in terms of log-sine integrals are contained in [12]. This was one of the motivations for a systematic study of log-sine integrals detailed in [13] and sketched next in Section 4.…”

Section: Multiple Mahler Measuresmentioning

confidence: 99%

“…The evaluations (73) and (82) together with (71) allow us to give polylogarithmic values for all multiple Mahler measures µ k (1 + x + y * ). A list of explicit values for the first six cases is given in [12].…”

Section: Lsmentioning

confidence: 99%

“…Many other Mahler measures studied in [12,4] were shown to have evaluations involving generalized log-sine integrals at π and π/3 as well. Several such examples are reviewed in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Mahler measures, short walks and log-sine integrals

Borwein

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

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The Mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years -very few closed forms are proven but more are conjectured. In the case of multiple Mahler measures more tractable but interesting families exist. Using values of log-sine integrals we provide systematic evaluations of various higher and multiple Mahler measures.The evaluations in terms of log-sine integrals become particularly useful in light of the fact that log-sine integrals may be automatically reexpressed as polylogarithmic values. We present this correspondence along with related generating functions for log-sine integrals.Our initial interest in considering Mahler measures stems from a study of uniform random walks in the plane as first introduced by Pearson. The main results on the moments of the distance traveled by an n-step walk, as well as the corresponding probability density functions, are reviewed. It is the derivative values of the moments that are Mahler measures.This work would be impossible without very extensive symbolic and numeric computations. It also makes frequent use of the new NIST Handbook of Mathematical Functions and similar tools. Our intention is to show off the interplay between numeric and symbolic computing while exploring the three mathematical topics in the title.

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“…More generally, as shown in [12], immediately from (70) and the definition (6) of the log-sine integrals, we have…”

Section: Multiple Mahler Measuresmentioning

confidence: 85%

“…, 1 + x + y k ) (5) and provided an evaluation of µ 2 (1+x+y * ). As observed in [12] this Mahler measure can be conveniently expressed in terms of log-sine integrals. Namely, if…”

Section: Introductionmentioning

confidence: 94%

Section: Multiple Mahler Measuresmentioning

confidence: 99%

Section: Lsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mahler measures, short walks and log-sine integrals

Borwein

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

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On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant

Pardo

2016

AAECC

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authors introduced zeta Mahler measure functions for multivariate polynomials [Cassaigne and Maillot (J Number Theory 83:226-255, 2000) called them "zeta Igusa" functions, but we follow here the terminology of Akatsuka (J Number Theory 129:2713-2734]. We generalize this notion by defining a zeta Mahler measure function Z X (·, f ) : C −→ C, where X is a compact probability space and f : X −→ C is a function bounded almost everywhere in X . We give sufficient conditions that imply that this function is holomorphic in certain domains. Zeta Mahler measure functions contains big amounts of information about the expected behavior of f on X . This generalization is motivated by the study of several quantities related to numerical methods that solve systems of multivariate polynomial equations. We study the functions Z(·, 1 / · aff ), Z(·, 1 /μ norm ) and Z(·, JAC), respectively associated to the norm of the affine zeros ( · aff ), the non-linear condition number (μ norm ) and the Jacobian determinant (JAC) of complete intersection zero-dimensional projective varieties. We find the exact value of these functions in terms of Gamma functions and we also describe their respective domains of holomorphy in C. With the exact value of these zeta functions we can immediately prove and exhibit expectations of some average properties of zero-dimensional algebraic varieties. For instance, the exact knowledge of Z(·, 1 / · aff ) yields as a consequence that the expectation of the mean of the logarithm of the norms of the affine zeros of a random system of polynomial equations is one half of the nth harmonic number H n . Other conclusions are exhibited along the manuscript. Using these generalized zeta functions we exhibit the exact value of the arithmetic height of the hyper-surface known as the discriminant variety (roughly speaking the variety formed by all systems of equations having a singular zero).

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Log-sine-polylog integrals and alternating Euler sums

Mező

2019

Acta Math. Hungar.

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Log-Sine Evaluations of Mahler Measures

Cited by 17 publications

References 17 publications

Mahler measures, short walks and log-sine integrals

Mahler measures, short walks and log-sine integrals

On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant

Log-sine-polylog integrals and alternating Euler sums

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