Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n )

Lupu, Cezar; Orr, Derek

doi:10.1007/s11139-018-0081-0

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…Up to k = 16 we find the a k to be rational numbers. This is of note since most representations of the Apery constant ζ(3) which involve ζ(2k) and rational coefficients have an overall factor of π 2 [66], while this result implies the existence of a representation with an overall factor of π 3 .…”

Section: ⇡ 6⇡ 10⇡ 14⇡mentioning

confidence: 91%

Resurgence and renormalons in the one-dimensional Hubbard model

Mariño

Reis

2022

SciPost Phys.

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We use resurgent analysis to study non-perturbative aspects of the one-dimensional, multicomponent Hubbard model with an attractive interaction and arbitrary filling. In the two-component case, we show that the leading Borel singularity of the perturbative series for the ground-state energy is determined by the energy gap, as expected for superconducting systems. This singularity turns out to be of the renormalon type, and we identify a class of diagrams leading to the correct factorial growth. As a consequence of our analysis, we propose an explicit expression for the energy gap at weak coupling in the multi-component Hubbard model, at next-to-leading order in the coupling constant. In the two-component, half-filled case, we use the Bethe ansatz solution to determine the full trans-series for the ground state energy, and the exact form of its Stokes discontinuity.

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Section: ⇡ 6⇡ 10⇡ 14⇡mentioning

confidence: 91%

Resurgence and renormalons in the one-dimensional Hubbard model

Mariño

Reis

2022

SciPost Phys.

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“…x 2n ; (9.10) see, for example, equation (4.28) of [27] or Proposition 3.1 of the more readily accessible [28]. Differentiation in (9.5) gives S 4 (x) = C 3 (x) and C 3 (x) = −S 2 (x), so we may integrate (9.10) twice, term by term, using the initial values C 3 (0) = ζ(3) and S 4 (0) = 0, to get S 4 (x) = xζ( 3…”

Section: Brownian Motion Exiting a Right-angled Trianglementioning

confidence: 99%

Deposition, diffusion, and nucleation on an interval

Georgiou

Wade

2022

Ann. Appl. Probab.

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Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studied in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson.

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Series representations and asymptotically finite representations for the numbers ζ(2m + 1) and β(2m)

Weba

2022

Journal of Mathematical Analysis and Applications

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Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n )

Cited by 3 publications

References 23 publications

Resurgence and renormalons in the one-dimensional Hubbard model

Resurgence and renormalons in the one-dimensional Hubbard model

Deposition, diffusion, and nucleation on an interval

Series representations and asymptotically finite representations for the numbers ζ(2m + 1) and β(2m)

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Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n )

Cited by 3 publications

References 23 publications

Resurgence and renormalons in the one-dimensional Hubbard model

Resurgence and renormalons in the one-dimensional Hubbard model

Deposition, diffusion, and nucleation on an interval

Series representations and asymptotically finite representations for the numbers ζ(2m + 1) and β(2m)

Contact Info

Product

Resources

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Series representations and asymptotically finite representations for the numbers ζ(2m + 1) and β(2m)