A functional relation for L-functions of graphs equivalent to the Riemann Hypothesis for Dirichlet L-functions

Friedli, Fabien

doi:10.1016/j.jnt.2016.05.025

Cited by 10 publications

(5 citation statements)

References 11 publications

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“…This should be compared with the results obtained in [9] and [8] where similar formulas were obtained in the case of the standard Laplacian and spectral L-functions, respectively.…”

Section: Introductionsupporting

confidence: 62%

The bundle Laplacian on discrete tori

Friedli

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We prove an asymptotic formula for the determinant of the bundle Laplacian on discrete d-dimensional tori as the number of vertices tends to infinity. This determinant has a combinatorial interpretation in terms of cycle-rooted spanning forests. We also establish a relation (in the limit) between the spectral zeta function of a line bundle over a discrete torus, the spectral zeta function of the infinite graph Z d and the Epstein-Hurwitz zeta function. The latter can be viewed as the spectral zeta function of the twisted continuous torus which is the limit of the sequence of discrete tori.

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“…This should be compared with the results obtained in [9] and [8] where similar formulas were obtained in the case of the standard Laplacian and spectral L-functions, respectively.…”

Section: Introductionsupporting

confidence: 62%

The bundle Laplacian on discrete tori

Friedli

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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“…Our result fits the work [FrKa17] and [Fri16], where similar statements are proved for the Riemann zeta function and certain Dirichlet L-functions. The case for α = 2 shows the most similarities with the statements in these previous works.…”

Section: Statement Of the Main Resultssupporting

confidence: 90%

“…Our goal is to prove an analogous result for our considerations of the Epstein zeta-function on a two-dimensional discrete torus. We use techniques of [FrKa17] and [Fri16] and define the function (using notation in Theorem 3.5)…”

Section: Epstein-riemann Conjecture and Discrete Zeta Functionmentioning

confidence: 99%

Spectral zeta function on discrete tori and Epstein-Riemann conjecture

Meiners¹,

Vertman²

2022

Preprint

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We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the α-dimensional torus. In the special case α = 1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. There, the authors have also formulated a conjecture on this asymptotics, that is equivalent to the Riemann conjecture. In this paper, inspired by the work of Friedli and Karlsson, we prove that a similar asymptotic expansion holds for α = 2. Similar argument applies to higher dimensions as well. A conjecture on this asymptotics gives an equivalent formulation of the Epstein-Riemann conjecture, if we replace the standard discrete Laplacian with the 9point star discrete Laplacian.

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“…One intriguing aspect that we mention is the functional equation of the type s vs 1 − s that appears also for these non-classical zeta functions. Notably one has ξ Z (1 − s) = ξ Z (s), see below, and the equivalence of certain asymptotic functional equations to the Riemann hypotheses for ζ(s) and certain Dirichlet L-functions [FK17,F16]. Although there are a few instances in the literature where such function are introduced for graphs, it seems that the first more systematic effort to study spectral zeta functions of graphs appear in my paper with Friedli [FK17].…”

Section: Introductionmentioning

confidence: 96%

Spectral zeta functions

Karlsson

2019

Preprint

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This paper discusses the simplest examples of spectral zeta functions, especially those associated with graphs, a subject which has not been much studied. The analogy and the similar structure of these functions, such as their parallel definition in terms of the heat kernel and their functional equations, are emphasized. Another theme is to point out various contexts in which these non-classical zeta functions appear. This includes Eisenstein series, the Langlands program, Verlinde formulas, Riemann hypotheses, Catalan numbers, Dedekind sums, and hypergeometric functions. Several open-ended problems are suggested with the hope of stimulating further research.

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A functional relation for L-functions of graphs equivalent to the Riemann Hypothesis for Dirichlet L-functions

Cited by 10 publications

References 11 publications

The bundle Laplacian on discrete tori

The bundle Laplacian on discrete tori

Spectral zeta function on discrete tori and Epstein-Riemann conjecture

Spectral zeta functions

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