The Lerch zeta function IV. Hecke operators

Lagarias, Jeffrey C.; Li, Wen Ching Winnie

doi:10.1186/s40687-016-0082-9

Cited by 4 publications

(4 citation statements)

References 38 publications

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“…It also bears the name of Lerch [19], who showed that for ℑz > 0 and q ∈ (0, 1) the functional equation ζ(z, q; 1 − s) = (2π) −s Γ(s) e( 1 4 s − zq)ζ(−q, z; s) + e(− 1 4 s + zq)ζ(q, 1 − z; s) holds; this is called Lerch's transformation formula. For an interesting account of the analytic properties of the Lipschitz-Lerch zeta function and related functions, we refer the reader to the work of Lagarius and Li [15][16][17][18]; see also Apostol [2].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On certain zeta functions associated with Beatty sequences

Banks

2018

Acta Arith.

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Let α > 1 be an irrational number of finite type τ . In this paper, we introduce and study a zeta function Z ♯ α (r, q; s) that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the Beatty sequence B(α) . .= (⌊αm⌋) m∈N . If r is an element of the lattice Z + Zα −1 , then Z ♯ α (r, q; s) continues analytically to the half-plane {σ > −1/τ } with its only singularity being a simple pole at s = 1. If r ∈ Z + Zα −1 , then Z ♯ α (r, q; s) extends analytically to the half-plane {σ > 1 − 1/(2τ 2 )} and has no singularity in that region.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On certain zeta functions associated with Beatty sequences

Banks

2018

Acta Arith.

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“…Φ(s, z, c)) is a simultaneous eigenfunction, acting on various function spaces. Part IV ( [47]) of this series considers such operators on a function space with real variables. These additional discrete symmetries together with the differential equation (1.12) suggest that there should be an automorphic interpretation of the Lerch zeta function, made in terms of the related functions L ± (s, a, c).…”

Section: Further Directionsmentioning

confidence: 99%

The Lerch Zeta function III. Polylogarithms and special values

Lagarias

2016

Res Math Sci

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This paper studies algebraic and analytic structures associated with the Lerch zeta function, complex variables viewpoint taken in part II. The Lerch transcendent (s, z, c) := ∞ n=0 z n (n+c) s is obtained from the Lerch zeta function ζ (s, a, c) by the change of variable z = e 2πia . We show that it analytically continues to a maximal domain of holomorphy in three complex variables (s, z, c), as a multivalued function defined over the base manifold C × (P 1 (C) {0, 1, ∞}) × (C Z) and compute the monodromy functions describing the multivaluedness. For positive integer values s = m and c = 1 this function is closely related to the classical m-th order polylogarithm Li m (z). We study its behavior as a function of two variables (z, c) for "special values" where s = m is an integer. For m ≥ 1 we show that it is a one-parameter deformation of Li m (z), which satisfies a linear ODE, depending on c ∈ C, of order m + 1 of Fuchsian type on the Riemann sphere. We determine the associated (m + 1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of Li m (z).

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“…e.g. [35]. In the last few decades, the most fundamental and influential works related to the Lerch zeta-function are [16], [43], [47], and [71], which are partly incorporated in [10].…”

Section: Introductionmentioning

confidence: 99%

The Boundary Lerch Zeta-Function and Short Character Sums à la Y. Yamamoto

2020

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As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185-195), there have appeared many instances in which only the imaginary part-the odd part-of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q-expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann's fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628-634). We may thus refer to this as the 'Fourier series-boundary q-series', and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275-289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function-and, a fortiori, the modular relation (S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014).

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The Lerch zeta function IV. Hecke operators

Cited by 4 publications

References 38 publications

On certain zeta functions associated with Beatty sequences

On certain zeta functions associated with Beatty sequences

The Lerch Zeta function III. Polylogarithms and special values

The Boundary Lerch Zeta-Function and Short Character Sums à la Y. Yamamoto

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