Asymptotic behavior of the Lerch transcendent function

Navas, Luis M.; Ruiz, Francisco J.; Malumbres, Juan

doi:10.1016/j.jat.2012.03.006

Cited by 7 publications

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“…In this manuscript, we give a precise proof that chains of truncated Beta distributions as defined in Section 2 converge to Benford's law. For the most part, the proof is quite straightforward while the convergence part is proved via a recent convergence result on Lerch transcendent [5]. This stresses the relationship between convergence to Benford's law and convergence of certain special functions.…”

Section: Introductionmentioning

confidence: 90%

Chains of Truncated Beta Distributions and Benford’s Law

Ruankong

Sumetkijakan

2019

Uniform Distribution Theory

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Section: Introductionmentioning

confidence: 90%

Chains of Truncated Beta Distributions and Benford’s Law

Ruankong

Sumetkijakan

2019

Uniform Distribution Theory

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“…Namely, one can obtain asymptotic series with simple explicit bounds on the error term, and use them to understand the growth of the functions and also certain limiting oscillatory phenomena brought to light by the Fourier expansions. We have done this for the classical Bernoulli and Euler polynomials in [8], for the Apostol-Bernoulli and Apostol-Euler polynomials in [10], and for the Lerch transcendent in [11]. Clearly one will have analogous results for the conjugate functions in each case.…”

Section: Some Consequences Derived From the Fourier Seriesmentioning

confidence: 98%

“…Since g(0) = g(1), g is a continuous and piecewise C 1 function that, by Dirichlet's theorem, is equal to the sum of its Fourier series. Moreover, the calculation of the Fourier coefficients is immediate (see the reasoning in Section 1 and [11] for additional details), and so we get…”

Section: Introductionmentioning

confidence: 99%

“…The relationships between (1.1) and Fourier series like (1.2) are well known and usually proved via complex analytic methods involving contour integrals. In previous papers (see [11,12]), we have derived some properties of the Lerch function based on Fourier series by direct calculation of the Fourier coefficients. Observe, for instance, the simplicity of the following reasoning.…”

Section: Introductionmentioning

confidence: 99%

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The Lerch transcendent from the point of view of Fourier analysis

Navas

Ruiz

Malumbres

2015

Journal of Mathematical Analysis and Applications

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We obtain some well-known expansions for the Lerch transcendent and the Hurwitz zeta function using elementary Fourier analytic methods. These Fourier series can be used to analytically continue the functions and prove the classical functional equations, which arise from the relations satisfied by the Fourier conjugate and flat Fourier series. In particular, the functional equation for the Riemann zeta function can be obtained in this way without contour integrals. The conjugate series for special values of the parameters yields analogous results for the Bernoulli and Apostol-Bernoulli polynomials. Finally, we give some consequences derived from the Fourier series.

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“…The result has been reproved often by various means. See for example [8], for the traditional approach using complex analytic techniques, or [14], for a short proof using basic Fourier Analysis.…”

Section: Introductionmentioning

confidence: 99%

Some functional relations derived from the Lindelöf-Wirtinger expansion of the Lerch transcendent function

2014

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The Lindelöf-Wirtinger expansion of the Lerch transcendent function implies, as a limiting case, Hurwitz's formula for the eponymous zeta function. A generalized form of Möbius inversion applies to the Lindelöf-Wirtinger expansion and also implies an inversion formula for the Hurwitz zeta function as a limiting case. The inverted formulas involve the dynamical system of rotations of the circle and yield an arithmetical functional equation.see [8, §1.11, p. 27] or [2, §25.14], for example. Logarithms and complex powers are always assumed to be principal. If |λ| < 1, then for any s ∈ C, the series converges uniformly in z over C\(−∞, 0], thus defining a holomorphic function of z in this region. If |λ| = 1, then the series converges in this same region provided Re s > 1. The value λ = 0 is considered trivial since it yields Φ(0, s, z) = z −s , and thus is usually excluded. There are multiple ways of defining analytic continuations of Φ in each parameter. This function, defined by Mathias Lerch in 1887 in his paper [11], includes as special cases of the parameters the Hurwitz and Riemann zeta functions and the polylogarithms, among others, and has applications ranging from number theory to physics. It is often used to obtain functional identities; see, for instance, [3,7,9,15].One often extends the domain to z ∈ C \ {0, −1, −2, . . . } by including the branch discontinuity of the principal argument. In addition, in this paper we shall only be considering s ∈ C with Re s < 0, in which case the summand (k + z) −s (with k ∈ N) in (1.1) continuously extends to z = −k by defining 0 −s = 0. Thus for Re s < 0 we can define Φ(λ, s, z) for all z ∈ C. For the parameter s, we shall often denote σ = Re s. Also, we shall use the notation C * = C \ {0}.

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Asymptotic behavior of the Lerch transcendent function

Cited by 7 publications

References 8 publications

Chains of Truncated Beta Distributions and Benford’s Law

Chains of Truncated Beta Distributions and Benford’s Law

The Lerch transcendent from the point of view of Fourier analysis

Some functional relations derived from the Lindelöf-Wirtinger expansion of the Lerch transcendent function

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