Riemann-Siegel integral formula for the Lerch zeta function

Balanzario, Eugenio P.; Sanchez-Ortiz, Jorge

doi:10.1090/s0025-5718-2011-02566-4

Cited by 4 publications

(5 citation statements)

References 9 publications

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“…where contour H is shown in Figure 1. In particular, this contour has a negative sign off compared to the same contour defined in Balanzario [11]. Using the relations…”

Section: Lerch Zeta Functionmentioning

confidence: 85%

“…It has a computational complexity of t 1/2 [6]. There are similar methods for the computation of the Lerch zeta [11] and Dirichlet L-function [12] as well. The formula is given by (1)…”

Section: Riemann Zeta Functionmentioning

confidence: 99%

“…We believe the extension of the formulas to complex a and λ should be straightforward. Following Balanzario [11], we can obtain a representation of the Lerch function valid in the entire complex plane for σ. We provide an easy derivation of this representation.…”

Section: Lerch Zeta Functionmentioning

confidence: 99%

“…However, due to immense interest in L-functions and the Generalized Riemann hypothesis, an efficient evaluation of the Dirichlet L-function is highly desired. Although the RS type formulas for Lerch zeta [11] and Dirichlet L-functions [12] do exist, their application to evaluating these functions to high accuracy has not been attempted. The error estimation with the RS formulas is very complex, even for the RZ function.…”

Section: Introductionmentioning

confidence: 99%

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Double Exponential method for Riemann Zeta, Lerch and Dirichlet L-functions

Tyagi¹

2022

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We provide efficient methods to evaluate the Riemann zeta, the Lerch zeta and the Dirichlet L-functions. The method uses the Riemann-Siegel (RS) type formulas and a modified double exponential (MDE) quadrature method near the saddle point of appropriate integrands. We provide simplified derivations of the RS formulas, containing finite series sums and residual integrals, for the Lerch and the Dirichlet L-functions. The MDE method allows us to remove the contribution of the singularities near the contour of integration for the residual integrals. The method allows us to evaluate the residual integrals to any prescribed accuracy. The numerical cost of evaluating them is minimal compared to the main series sums. Thus even highly oscillatory integrands for these functions can be evaluated with the same complexity as the RS formula. In particular, the numerical complexity to evaluate the ζ(s) and the Lerch zeta function with s = σ + it scales as √ t, and for Dirichlet L-function it scales as max(q, √ qt). The method allows the automatic setting of integral cutoffs and finite discretization size to achieve prescribed accuracy. Furthermore, it ensures that every halving of the discretization interval almost leads to doubling the accuracy of the results. Thus, it allows for a further check on the accuracy of the results obtained.

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“…where contour H is shown in Figure 1. In particular, this contour has a negative sign off compared to the same contour defined in Balanzario [11]. Using the relations…”

Section: Lerch Zeta Functionmentioning

confidence: 85%

“…It has a computational complexity of t 1/2 [6]. There are similar methods for the computation of the Lerch zeta [11] and Dirichlet L-function [12] as well. The formula is given by (1)…”

Section: Riemann Zeta Functionmentioning

confidence: 99%

Section: Lerch Zeta Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Double Exponential method for Riemann Zeta, Lerch and Dirichlet L-functions

Tyagi¹

2022

Preprint

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“…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, . .…”

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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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