Spectral Zeta Functions

Karlsson, Anders

doi:10.1007/978-3-030-44097-8_10

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…Special values of ζ Z/nZ (s) and ζ(s). Sums of powers of the sine function are special cases of important sums in works of Dedekind, Verlinde, Dowker and others in number theory and physics; see [Do92,Z96,Do15,K20]. From our perspective, and also partly from Dowker's, they are special values of spectral zeta functions of discrete circles.…”

Section: Proof Of the Volume And Product Formulasmentioning

confidence: 87%

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Pallich

2023

Acta Arith.

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Section: Proof Of the Volume And Product Formulasmentioning

confidence: 87%

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Pallich

2023

Acta Arith.

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“…defined for any complex s that makes sense, as in the Riemann zeta-function corresponding to the sum of the integers λ n = n over n = 1, 2, .., +∞. Such spectral zeta-functions have been much studied, in particular for Laplacians on compact manifolds and for Schrödinger operators with an infinite discrete spectrum N = +∞ (see the reviews [93][94][95] and references therein) together with the spectral partition function…”

Section: Appendix D: Parametrization Of One-dimensional Diffusion Pro...mentioning

confidence: 99%

On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival

Mazzolo¹,

Monthus²

2023

Preprint

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For continuous-time ergodic Markov processes, the Kemeny time τ * is the characteristic time needed to converge towards the steady state P * (x) : in real-space, the Kemeny time τ * corresponds to the average of the Mean-First-Passage-Time τ (x, x0) over the final configuration x drawn with the steady state P * (x), which turns out to be independent of the initial configuration x0; in the spectral domain, the Kemeny time τ * corresponds to the sum of the inverses of all the non-vanishing eigenvalues λn = 0 of the opposite generator. We describe many illustrative examples involving jumps and/or diffusion in one dimension, where the Kemeny time can be explicitly computed as a function of the system-size, via its real-space definition and/or via its spectral definition : we consider both reversible processes satisfying detailed-balance where the eigenvalues are real, and irreversible processes characterized by non-vanishing steady currents where the eigenvalues can be complex. In particular, we study the specific properties of the Kemeny times for Markov processes with stochastic resetting, and for absorbing Markov processes conditioned to survive forever. I. * τ * (x 0 ) ≡ N x=1 P * (x)τ (x, x 0 ) ≡ τ Space * independent of x 0(2)

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“…This function appears (see e.g. [Kar19]) in several places in the literature concerning Eisenstein series. Our formula also links to the famous Catalan numbers Sta15]).…”

Section: Introductionmentioning

confidence: 95%

Zeta functions of graphs, their symmetries and extended Catalan numbers

Dubout

2019

Preprint

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In this paper we study spectral zeta functions associated to finite and infinite graphs. First we establish a meromorphic continuation of these functions under some general conditions. Then we study special values in the case of standard lattice graphs associated to free abelian groups. In particular we connect it to Catalan numbers in several ways, and obtain some non-trivial special values and functional symmetries. Furthermore we relate the values at the negative integers with the more studied Ihara zeta functions, and prove a few minor results that seem not to have been recorded before. Finally we consider the characteristic polynomial of the graph Laplacians, and in particular completely determine its coefficients for cyclic graphs using new analytical methods.

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Spectral Zeta Functions

Cited by 5 publications

References 23 publications

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival

Zeta functions of graphs, their symmetries and extended Catalan numbers

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