A note on the gaps between zeros of Epstein's zeta-functions on the critical line

Baier, Stephan; Kotyada, Srinivas; Sangale, Usha K

doi:10.7169/facm/1630

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Potter and Tichmarsh [17] showed N Ep (T ) ≫ T 1/2−ε , Sankaranarayanan's result [18] is N Ep (T ) ≫ T 1/2 / log T , Jutila and Srinivas [14] proved N Ep (T ) ≫ T 5/11−ε . The current (October, 2019) best result is N Ep (T ) ≫ T 4/7−ε , shown by Baier, Srinivas and Sangale [2].…”

Section: 2mentioning

confidence: 95%

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Functional equation and zeros on the critical line of the quadrilateral zeta function

Nakamura

2019

Preprint

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Let 0 < a ≤ 1/2 and define the quadrilateral zeta function by Q(s, a) := ζ(s, a) + ζ(s, 1 − a) + Li(s, a) + Li(s, 1 − a), where ζ(s, a) is the Hurwitz zeta function and Li(s, a) is the periodic zeta function.In the present paper, we prove that for any 0 < a ≤ 1/2, there are positive constants A(a) and T 0 (a) such that the numbers of zeros of Q(s, a) on the line segment from 1/2 to 1/2 + iT is at least A(a)T whenever T ≥ T 0 (a). We also give some remarks on the functional equations of zeta functions related to Hamburger's and Knopp's theorems.

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Section: 2mentioning

confidence: 95%

“…Corollary 1.4. The function Q(2s, a) satisfies the conditions (1), (2) and -(3a')-The both functions φ(s) and φ(s/2) can be expanded in a general Dirichlet series converge in some half-plane.…”

Section: 2mentioning

confidence: 99%

Functional equation and zeros on the critical line of the quadrilateral zeta function

Nakamura

2019

Preprint

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“…Potter and Tichmarsh [13] showed N CL Ep (T ) ≫ T 1/2−ε . The current (January, 2020) best result is N CL Ep (T ) ≫ T 4/7−ε , shown by Baier, Srinivas and Sangale [2].…”

mentioning

confidence: 96%

On Lerch's formula and zeros of the quadrilateral zeta function

Nakamura¹

2020

Preprint

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Let 0 < a ≤ 1/2 and define the quadrilateral zeta function by 2Q(s, a) := ζ(s, a)+ ζ(s, 1 − a)+ Li s (e 2πia )+ Li s (e 2πi(1−a) ), where ζ(s, a) is the Hurwitz zeta function and Li s (e 2πia ) is the periodic zeta function.In the present paper, we show that there exists a unique real number a 0 ∈ (0, 1/2) such that Q(σ, a 0 ) has a unique double real zero at σ = 1/2 when σ ∈ (0, 1), for any a ∈ (a 0 , 1/2], the function Q(σ, a) has no zero in the open interval σ ∈ (0, 1) and for any a ∈ (0, a 0 ), the function Q(σ, a) has at least two real zeros in σ ∈ (0, 1).Moreover, we prove that Q(s, a) has infinitely complex zeros in the region of absolute convergence and the critical strip when a ∈ Q ∩ (0, 1/2) \ {1/6, 1/4, 1/3}. The Riemannvon Mangoldt formula for Q(s, a) is also shown.Theorem A. All real zeros of ζ(s) are simple and at only the negative even integers.

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Generation of off-critical zeros for hypercubic Epstein zeta functions

Travěnec

Šamaj

2022

Applied Mathematics and Computation

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A note on the gaps between zeros of Epstein's zeta-functions on the critical line

Cited by 4 publications

References 8 publications

Functional equation and zeros on the critical line of the quadrilateral zeta function

Functional equation and zeros on the critical line of the quadrilateral zeta function

On Lerch's formula and zeros of the quadrilateral zeta function

Generation of off-critical zeros for hypercubic Epstein zeta functions

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