Double tails of multiple zeta values

Akhilesh, P.

doi:10.1016/j.jnt.2016.06.020

Cited by 10 publications

(19 citation statements)

References 3 publications

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“…. , k d ) ∈ N d with k 1 2, the multiple zeta value ζ(k) is defined by the following infinite series ζ(k) = ζ(k 1 , . .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Hence some topology of B(0, 1) is needed. In fact, B(0, 1) is a complete normed space with the norm given by f = sup q∈(0, 1) |f (q)|, ∀f ∈ B(0, 1).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Then the facts Since lim n→∞ ζ(n + 2, 1) = 0, we get lim n→∞ ζ[3, {1} n ) = 0, which implies that 0 ∈ QZZ (1) . Similarly, let k = (k 1 , .…”

Section: Proof We Have Limmentioning

confidence: 89%

“…Proof of Theorem 1.3. To prove Theorem 1.3, we recall the concept of double tails of multiple zeta values of Akhilesh from [1]. Let k = (k 1 , .…”

Section: 3mentioning

confidence: 99%

“…Conversely, for any f ∈ QZZ (1) , there exists a sequence (ζ[k(n), r(n))) n∈N such that k(n) is admissible, r(n) ∈ N and lim n→∞ ζ[k(n), r(n)) = f.…”

Section: Proof We Have Limmentioning

confidence: 99%

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Topological properties of q-analogues of multiple zeta values

Pan

2019

Int. J. Number Theory

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In the space of bounded real-valued functions on the interval (0, 1), we study the convergent sequences of q-analogues of multiple zeta values which do not converge to 0. And we obtain the derived sets of the set of some qanalogue of multiple zeta values.This q-analogue was first studied by Bradley [2] and independently by Zhao [4]. Here we introduce another q-analogue of multiple zeta values. Let r ∈ N, then we.( 1.2) Different from multiple zeta values, the multiple q-zeta values have a parameter q. Hence we work in the function space B(0, 1), which is the set of bounded realvalued functions on the open interval (0, 1). Since the multiple q-zeta values we consider here belong to B(0, 1) (see Remark 2.4), we just study the following two subspaces of B(0, 1):We define an order of B(0, 1) as follows. Let f, g ∈ B(0, 1). The function f is smaller than g, if f (q) < g(q) for any q ∈ (0, 1). We denote this by f < g. Then we can find the maximum element of QZ. Theorem 1.1. For any admissible multi-index k, we have ζ[k] ζ[2]. In other words, ζ[2] is the maximum element of QZ. While for the subspace QZZ, we only obtain an upper bound. Theorem 1.2. For any admissible multi-index k and any r ∈ N, we have ζ[k, r) < ζ[2]. In other words, ζ[2] is an upper bound of QZZ.

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“…. , k d ) ∈ N d with k 1 2, the multiple zeta value ζ(k) is defined by the following infinite series ζ(k) = ζ(k 1 , . .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Hence some topology of B(0, 1) is needed. In fact, B(0, 1) is a complete normed space with the norm given by f = sup q∈(0, 1) |f (q)|, ∀f ∈ B(0, 1).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Then the facts Since lim n→∞ ζ(n + 2, 1) = 0, we get lim n→∞ ζ[3, {1} n ) = 0, which implies that 0 ∈ QZZ (1) . Similarly, let k = (k 1 , .…”

Section: Proof We Have Limmentioning

confidence: 89%

“…Proof of Theorem 1.3. To prove Theorem 1.3, we recall the concept of double tails of multiple zeta values of Akhilesh from [1]. Let k = (k 1 , .…”

Section: 3mentioning

confidence: 99%

“…Conversely, for any f ∈ QZZ (1) , there exists a sequence (ζ[k(n), r(n))) n∈N such that k(n) is admissible, r(n) ∈ N and lim n→∞ ζ[k(n), r(n)) = f.…”

Section: Proof We Have Limmentioning

confidence: 99%

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