Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume

Jay Jorgenson; Lejla Smajlovic; Polyxeni Spilioti

Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume

T0 review · 0 major / 5 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Normalized log spectral determinants of weighted Laplacians on growing-genus hyperbolic surfaces converge to a constant fixed only by the limiting weight.

desk verdict Solid deterministic extension of Naud’s large-genus det limit to weighted Maass–Laplacians, with a usable diameter-free PGT; conditional on three clean hypotheses that match the random-surface literature. read the letter →

arxiv 2607.26779 v1 pith:JWTS7WVH submitted 2026-07-29 math.SP math.NT

Jay Jorgenson , Lejla Smajlovic , Polyxeni Spilioti This is my paper · ORCID

classification math.SPmath.NT MSC 58J5011M3611F7230F10

keywords spectraldeterminantweightedLaplacianprimegeodesictheoremSelbergzetafunctioncompactRiemannsurfaceslargegenusasymptoticsmultipliersystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves two linked results for compact hyperbolic surfaces carrying unitary multiplier systems of admissible weight. First it gives an explicit prime-geodesic theorem for the weighted counting function, with an error whose constants depend only on genus, multiplier dimension, systole and the lowest nonzero eigenvalues of the weighted and ordinary Laplacians. Second, for any sequence of such surfaces whose genus tends to infinity, under three mild geometric hypotheses (a weak spectral gap, uniform lower bound on the systole, and controlled non-accumulation of short geodesics), the quantity log det(Δ_{2k_n}) / vol(X_n) converges to an explicit constant C_α that depends solely on the limiting weight α = lim k_n. The result is deterministic yet matches the predictions of the three standard random-surface models, and it recovers earlier unweighted theorems as the special case α = 0.

What carries the argument

An explicit weighted prime-geodesic theorem (Theorem A) that expresses the Chebyshev-type function Ψ(x,χ) as a sum over small eigenvalues plus an error of size O(x^{3/4}) or O(x^{5/6}) with fully effective constants; this asymptotic is converted via Stieltjes integration into control of the special value of the Selberg zeta at s = 1, which in turn equals the regularized determinant up to elementary factors.

What would settle it

Exhibit a sequence of compact hyperbolic surfaces of unbounded genus that satisfy uniform discreteness and non-accumulation of short geodesics, yet for which N_{k,n}/(λ_{1,n} vol(X_n)) tends to infinity, and check whether log det(Δ_{2k_n})/vol(X_n) still converges to C_α.

Watch

Extended reading notes

Core claim

For sequences of compact hyperbolic surfaces of unbounded genus equipped with m-dimensional unitary multipliers of weights 2k_n → 2α ∈ [0,2), the normalized spectral determinant satisfies |2π log det(Δ_{2k_n}) / (m vol(X_n)) − C_α| < ε for all large n, where C_α is the explicit combination of zeta values, Gamma values and Barnes G-values given in (5.5), provided the surfaces obey a weak spectral gap, uniform discreteness of the Fuchsian group, and hypothesis H(C,L_ε,κ).

Load-bearing premise

The number of eigenvalues at most 1/4 must not grow faster than the product of the first positive eigenvalue and the volume; if that weak gap fails the tail integral that produces the limit diverges.

Editorial extensions

If this is right

When the limiting weight is zero the constant collapses to the classical value 2ζ'(−1)+log√(2π)−1/4, recovering the unweighted determinant asymptotics.
For any fixed rational weight p/q the same limit holds along the arithmetic progression of genera congruent to 1 mod q.
The magnetic Laplacian (weight equal to the quantized field strength) inherits the same volume-normalized determinant limit.
Any deterministic sequence obeying the three geometric hypotheses automatically satisfies the same determinant law previously known only in probabilistic models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same Stieltjes-integral argument should extend, with only minor changes, to the determinants of higher powers of the weighted Laplacian or to Ruelle zeta values at other integer points.
If a uniform spectral gap stronger than the weak gap can be proved for a concrete arithmetic tower, the error term in the determinant convergence becomes completely effective.
The explicit dependence of the prime-geodesic error on the systole suggests that surfaces with very short geodesics will exhibit slower approach to C_α, a prediction that can be tested numerically on random covers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The punchline is straightforward: they get a deterministic volume-normalized limit for log det of the weighted Laplacian under three explicit hypotheses (weak spectral gap, uniform discreteness, non-accumulation of short geodesics), and the constant C_α depends only on the limiting weight α. This is the natural weighted/m-dimensional extension of Naud 2023, and they also supply the diameter-independent weighted prime-geodesic theorem needed to make the volume asymptotics work.

What is new is exactly that pair. Theorem A gives an explicit error for Ψ(x,χ) controlled by genus, m, systole and the first eigenvalues of Δ_{2k} and Δ_0; the diameter is gone, which is the point. Theorem B then feeds the uniform version (Theorem 4.1) into the Gon determinant–Selberg identity, writes log Z as a Stieltjes integral of Ψ, splits at an ε-dependent L_ε, kills the tail by the weak-gap hypothesis and the compact piece by H(C,L_ε,κ). The constant (5.5) comes cleanly from the identity contribution and Barnes G; nothing is fitted. The three hypotheses are flagged, compared with WP/Brooks–Makover/random covers, and used only where needed. Corollaries for fixed rational weight and for α=1/2 are immediate and useful.

Soft spots are minor and proportional. The weak-gap assumption is external; if N_{k,n} grows faster than λ_1 vol the tail estimate fails. They say so, and note it is weaker than a uniform gap and expected in the random models. The PGT constants are not optimized, but they never claim they are. No circularity, no hidden diameter dependence, no unjustified interchange. The stress-test found nothing load-bearing wrong, and I agree.

This is for people already working on spectral geometry of large-genus surfaces or random hyperbolic surfaces who want a deterministic benchmark and the weighted case. The math is classical analytic number theory done carefully; a serious referee can check it line-by-line. I would accept for peer review, cite the limit and the constant when I need the weighted statement, and bring the PGT + determinant sections to reading group.

Referee Report

0 major / 5 minor

Summary. The paper proves two results for compact hyperbolic surfaces with unitary multiplier systems of admissible weight. Theorem A gives an explicit prime-geodesic theorem for the weighted counting function Ψ(x,χ), with two forms of the error: one depending on genus, systole, and small eigenvalues of Δ_{2k} and Δ_0 (via Randol-type test functions and an effective Weyl law), and a uniform version whose error depends only on genus, dimension, and systole (via smoothed cutoffs in the style of Wu–Xue). Theorem B then uses the uniform PGT, the Gon determinant–Selberg identity, and a Stieltjes representation of log Z(s;χ) to show that, for sequences of surfaces of genus tending to infinity satisfying a weak spectral gap, uniform discreteness of the Fuchsian group, and Naud’s non-accumulation hypothesis H(C,L_ε,κ), one has 2π log det(Δ_{2k_n})/(m Vol(X_n)) → C_α, where C_α is an explicit constant built from ζ'(-1), Γ, and the Barnes G-function and depends only on α = lim k_n. The result is deterministic and is shown to be compatible with the Weil–Petersson, Brooks–Makover, and random-cover models.

Significance. The work supplies a deterministic large-genus limit for spectral determinants of weighted (and magnetic) Laplacians under hypotheses weaker than a uniform spectral gap, extending Naud’s k=0, m=1 theorem and covering new admissible-weight regimes (e.g., fixed rational weight on genera congruent to 1 mod q). The explicit, diameter-free error terms in the weighted PGT are of independent interest and are precisely what make the volume-normalized determinant asymptotics accessible. Constants are tracked effectively, the reduction via the Selberg zeta and Barnes G-function is standard and transparent, and compatibility with the three main random-surface models is carefully documented. These are solid, usable contributions to spectral geometry and the arithmetic of Fuchsian groups.

minor comments (5)

[§1–§2, Theorem B] Notation for the weight is slightly inconsistent across the abstract, Theorem B, and §2–§5 (k versus 2k, and the range kn ∈ (0,2) versus α ∈ [0,1)). A single convention stated once in §2 and used uniformly would help.
[§4, after (4.9)] In the proof of Theorem 4.1 the universal threshold is taken as x_0 = 10^{18}. It is fine as an existence statement, but a brief remark that any sufficiently large absolute constant works (or a pointer to where the 0.001 cutoff can be relaxed) would reassure readers who only need the asymptotic regime.
[§5, Remark 5.1] Remark 5.1 mentions the Buser–Sarnak example and Cheeger-constant reformulations; a one-sentence cross-reference to the precise place where the weak-gap hypothesis enters the tail estimate (5.12)–(5.15) would make the logical dependence even clearer.
[§5, Corollaries 5.7–5.8] Corollaries 5.7–5.8 are attractive special cases; adding a short sentence on whether the same limit holds for sequences with kn → α along other admissible paths (not just constant or fixed rational weight) would round out the discussion.
[References] A few bibliographic items are still listed as preprints with arXiv numbers only (e.g., AM25, AM26, MN26, GK24). Update status or page ranges where available before final publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: conditional limit under external hypotheses, with C_α from known special-function identity terms

full rationale

The central claim (Theorem B / (1.7)) is a conditional deterministic asymptotic: under weak spectral gap (5.2), uniform discreteness (5.3), and H(C,L_ε,κ), log det(Δ_{2k_n})/(m vol(X_n)) approaches the explicit constant C_α in (5.5). The proof chain is: Gon95 identity (2.13) reduces det to Selberg Z(s;χ_n) plus identity factor Z_I; the s→1+ limit of log Z is the Stieltjes integral (5.11) of the weighted prime-counting Ψ; the uniform PGT (Theorem 4.1) plus weak gap controls the tail integral (5.12)–(5.15); H and uniform discreteness control the compact piece (5.16)–(5.17). C_α is assembled from the known Barnes G / Gamma factors in Z_I and the constant ˜c (2.14)–(2.15), not fitted to the sequence. The three geometric/spectral hypotheses are stated as external assumptions, not restated conclusions. Self-citations (e.g. FJK11 for explicit-error methods) supply background tools and are not uniqueness theorems that force the limit. No step reduces the claimed prediction to its inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central limit rests on three external geometric/spectral hypotheses plus the classical Selberg trace formula and the known relation between the zeta-regularized determinant and the Selberg zeta function. No parameters are fitted to data; C_α is an explicit combination of special values of Γ, G and ζ′(−1). No new physical or geometric entities are postulated.

assumptions (6)

standard math Selberg trace formula for unitary multiplier systems of admissible weight (Gon95 / Hej83 / Fis87 form used in (2.8)–(2.10))
Invoked throughout §§3–5 as the identity relating spectral and geometric sides; treated as established background.
standard math Relation det(Δ_{2k}−s(1−s)) = Z(s) Z_I(s) e^{c̃} for Re(s)>1 (Gon95, Thm 3 / (2.13))
Used in §5 to reduce the determinant asymptotic to the special value of log Z(s) at s=1.
domain assumption Weak spectral gap: lim sup N_{k,n}/(λ_{1,n} vol(X_n)) = β < ∞ (1.5)/(5.2)
Load-bearing external hypothesis needed to kill the integral tail (5.12)–(5.15); not proved for arbitrary sequences.
domain assumption Uniform discreteness: sys(X_n) ≥ log η > 0 for all n (1.6)/(5.3)
Ensures the lower limit of integration in (5.11) stays away from 1 uniformly; expected in random models but assumed here.
domain assumption Non-accumulation H(C,L_ε,κ): |{P : N(P)≤L_ε}| ≤ C vol(X_n)^κ (Nau23)
Controls the short-geodesic contribution (5.16)–(5.17); holds a.s. in the three random models but is an assumption for deterministic sequences.
domain assumption When α=0 the multiplicities m(λ_{0,n}) remain uniformly bounded in n
Extra hypothesis stated in Theorem B to handle the zero-weight case; used in the bound after (5.13).

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Pith. "Pith review of Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume." pith.science (2026). https://pith.science/paper/JWTS7WVH

@misc{pith2026260726779,
  author       = {Pith},
  title        = {Pith review of: Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWTS7WVH}},
  note         = {Machine review of arXiv:2607.26779}
}

abstract

Let $(X,\chi,k)$ be a triple consisting of a smooth, compact hyperbolic Riemann surface $X$ of genus $g$, and an $m$ dimensional unitary multiplier system $\chi$ of admissible weight $k$. Our first result establishes an analogue of the prime geodesic theorem for the weighted prime geodesic counting function associated to $(X,\chi,k)$. The error term we obtain is explicit with effectively computable constants which depend solely on the genus of $X$, the dimension of $\chi$, the length of shortest geodesic on $X$ and the smallest non-zero eigenvalues of the weighted Laplacian $\Delta_{2k}$ as well that of the scalar Laplacian $\Delta_{0}$. Our second result studies the asymptotic behavior of the spectral determinant $\det\Delta_{2k_n}$ for a sequence $(X_{n}, \chi_{n}, k_{n})$ for which the genus of $X_n$ tends to infinity. Under reasonably general circumstances, namely the existence of a weak spectral gap, a uniform discreteness of the underlying Fuchsian group, and a type of non-accumulation of bounded geodesics, we prove that $\log\det\Delta_{2k_n}/\mathrm{vol}(X_{n})$ converges to a constant $C_{\alpha}$ which depends only on $\alpha=\lim_{n\to\infty} k_n$. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.

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Reference graph

Works this paper leans on

213 extracted references · 24 canonical work pages

[1]

Turaev, V. G. , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 1986 , NUMBER =

1986
[2]

Journal of the Institute of Mathematics of Jussieu , author=

Twisted. Journal of the Institute of Mathematics of Jussieu , author=. 2025 , pages=

2025
[3]

Functional Analysis and Its Applications , volume=

Determinants of Cauchy-Riemann operators over a Riemann surface , author=. Functional Analysis and Its Applications , volume=. 1985 , publisher=

1985
[4]

An extension of a theorem by Cheeger and M

Bismut, Jean-Michel and Zhang, Weiping , journal=. An extension of a theorem by Cheeger and M
[5]

Geometry & Topology , volume=

Reidemeister torsion, complex volume and the Zograf infinite product for hyperbolic 3--manifolds , author=. Geometry & Topology , volume=. 2019 , publisher=

2019
[6]

Inventiones mathematicae , volume=

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds , author=. Inventiones mathematicae , volume=. 2022 , publisher=

2022
[7]

arXiv preprint arXiv:2303.00312 , year=

A Ruelle dynamical zeta function for equivariant flows , author=. arXiv preprint arXiv:2303.00312 , year=

arXiv
[8]

The twisted

B. The twisted. arXiv preprint arXiv:2110.06683 , year=

arXiv

Show all 213 references

[9]

Frahm, Jan and Spilioti, Polyxeni , journal=. Twisted. 2023 , publisher=

2023
[10]

International Mathematics Research Notices , volume =

Shen, Shu , title = ". International Mathematics Research Notices , volume =. 2021 , month =

2021
[11]

Ruelle zeta functions of hyperbolic manifolds and

M. Ruelle zeta functions of hyperbolic manifolds and. The Journal of Geometric Analysis , volume=. 2021 , publisher=

2021
[12]

Annals of Global Analysis and Geometry , volume=

Functional equations of Selberg and Ruelle zeta functions for non-unitary twists , author=. Annals of Global Analysis and Geometry , volume=. 2020 , publisher=

2020
[13]

Communications on Pure and Applied Mathematics , volume=

Selberg's trace formula as applied to a compact Riemann surface , author=. Communications on Pure and Applied Mathematics , volume=. 1972 , publisher=

1972
[14]

Annals of Mathematics , volume=

Closed geodesics and the -invariant , author=. Annals of Mathematics , volume=. 1978 , publisher=

1978
[15]

Pfaff, Jonathan and Raimbault, Jean , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/tran/7875 , URL =

2020 doi
[16]

Forum Mathematicum , year =

Anton Deitmar and Polyxeni Spilioti and Yasuro Gon , title =. Forum Mathematicum , year =
[17]

Inventiones mathematicae , volume=

Asymptotics of class numbers , author=. Inventiones mathematicae , volume=. 2005 , publisher=

2005
[18]

, TITLE =

Deitmar, A. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2004 , NUMBER =

2004
[19]

Lefschetz formulas for flows , author=. Contemp. Math , volume=
[20]

Analytic torsion and

M. Analytic torsion and. Advances in Mathematics , volume=. 1978 , publisher=

1978
[21]

Annals of Mathematics , volume=

Analytic torsion and the heat equation , author=. Annals of Mathematics , volume=. 1979 , publisher=

1979
[22]

1936 , Publisher =

Georges. 1936 , Publisher =

1936
[23]

Franz, Wolfgang , journal=
[24]

Homotopieringe und linsenr

Reidemeister, Kurt , booktitle=. Homotopieringe und linsenr. 1935 , organization=

1935
[25]

Complex powers of an elliptic operator , author=. Proc. Symp. Pure Math. , volume=
[26]

Dang, Nguyen Viet and Guillarmou, Colin and Riviere, Gabriel and Shen, Shu , journal =. The. 2019 , publisher =

2019
[27]

arXiv preprint arXiv:1911.09931 , year =

Dynamical torsion for contact Anosov flows , author =. arXiv preprint arXiv:1911.09931 , year =

1911 arXiv
[28]

arXiv preprint arXiv:1802.08886 , year =

Holomorphic torsion with coefficients and geometric zeta functions for certain Hermitian locally symmetric manifolds , author =. arXiv preprint arXiv:1802.08886 , year =

arXiv
[29]

Annals of Mathematics , pages =

Determination of the cobordism ring , author =. Annals of Mathematics , pages =. 1960 , publisher =

1960
[30]

1998 , publisher =

On Thom spectra, orientability, and cobordism , author =. 1998 , publisher =

1998
[31]

1975 , organization =

Atiyah, Michael F and Patodi, Vijay Kumar and Singer, IM , booktitle =. 1975 , organization =

1975
[32]

Selecta Mathematica , year =

Fedosova, Ksenia and Pohl, Anke , title =. Selecta Mathematica , year =
[33]

Bulletin of the American Mathematical Society , volume =

Whitehead torsion , author =. Bulletin of the American Mathematical Society , volume =. 1966 , publisher =

1966
[34]

International Mathematics Research Notices , volume =

A canonical quadratic form on the determinant line of a flat vector bundle , author =. International Mathematics Research Notices , volume =. 2008 , publisher =

2008
[35]

Heat kernels on covering spaces and topological invariants , url =

Lott, John , doi =. Heat kernels on covering spaces and topological invariants , url =. J. Differential Geom. , number =
[36]

2013 , publisher =

L2-invariants: theory and applications to geometry and K-theory , author =. 2013 , publisher =

2013
[37]

Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences , volume =

Complex-valued analytic torsion for flat bundles and for holomorphic bundles with (1, 1) connections , author =. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences , volume =. 2010 , publisher =

2010
[38]

The twisted

Fedosova, Ksenia , journal =. The twisted
[39]

Proceedings of the American Mathematical Society , volume =

Cappell-Miller analytic torsion for manifolds with boundary , author =. Proceedings of the American Mathematical Society , volume =
[40]

Analytic torsion and L^

Müller, Werner and Pfaff, Jonathan , journal =. Analytic torsion and L^. 2013 , publisher =

2013
[41]

Journal of the Institute of Mathematics of Jussieu , volume =

The asymptotic growth of torsion homology for arithmetic groups , author =. Journal of the Institute of Mathematics of Jussieu , volume =. 2013 , publisher =

2013
[42]

arXiv preprint arXiv:1709.00760 , year =

Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy , author =. arXiv preprint arXiv:1709.00760 , year =

arXiv
[43]

Deitmar , title =

A. Deitmar , title =. Advances in Mathematics , year =
[44]

Annals of Global Analysis and Geometry , year =

Spilioti, Polyxeni , title =. Annals of Global Analysis and Geometry , year =
[45]

The trace class conjecture in the theory of automorphic forms

M. The trace class conjecture in the theory of automorphic forms. II , journal =. 1998 , volume =

1998
[46]

Gangolli, R. , doi =. Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces , url =. Acta Math. , pages =
[47]

, title =

Gilkey, Peter B. , title =. Algebraic and differential topology---global differential geometry , series =. 1984 , mrclass =

1984
[48]

, title =

Taylor, M. , title =. 2011 , publisher =

2011
[49]

, title =

Taylor, M. , title =. 1981 , publisher =

1981
[50]

Spilioti, Polyxeni , journal =. Twisted
[51]

Ruelle and

Spilioti, Polyxeni , journal =. Ruelle and
[52]

Functional equations of

Spilioti, Polyxeni , journal =. Functional equations of
[53]

Fried, David , title =. Ann. Sci. \'Ecole Norm. Sup. (4) , fjournal =. 1986 , number =

1986
[54]

Dowker, Fay and Elizalde, Emilio and Kirsten, Klaus , title =. J. Phys. A , fjournal =. 2012 , number =

2012
[55]

Fucci, Guglielmo and Kirsten, Klaus , title =. Appl. Math. Comput. , fjournal =. 2013 , pages =

2013
[56]

Elizalde, Emilio and Kirsten, Klaus and Robles, Nicolas and Williams, Floyd , title =. Int. J. Geom. Methods Mod. Phys. , fjournal =. 2015 , number =

2015
[57]

Classical Quantum Gravity , fjournal =

Aurich, Ralf and Lustig, Sven and Steiner, Frank and Then, holger , title =. Classical Quantum Gravity , fjournal =. 2004 , number =

2004
[58]

Quine, J. R. and Heydari, S. H. and Song, R. Y. , title =. Trans. Amer. Math. Soc. , fjournal =. 1993 , number =

1993
[59]

Kac, Mark , title =. Amer. Math. Monthly , fjournal =. 1966 , number =

1966
[60]

1972 , pages =

Warner, Garth , title =. 1972 , pages =

1972
[61]

Notices Amer

Ruelle, David , title =. Notices Amer. Math. Soc. , fjournal =. 2002 , number =

2002
[62]

, title =

Arthur, James G. , title =. Representation theory and harmonic analysis on semisimple
[63]

Tu , title =

Raoul Bott and Loring W. Tu , title =. 1982 , howPublished =

1982
[64]

1972 , howPublished =

Garth. 1972 , howPublished =

1972
[65]

1995 , pages =

Bunke, Ulrich and Olbrich, Martin , title =. 1995 , pages =

1995
[66]

1992 , publisher =

Nicole Berline and Ezra Getzler and Mich\`ele Vergne , title =. 1992 , publisher =

1992
[67]

Differential and

Bott, Raoul , title =. Differential and
[68]

Atiyah, M. F. and Patodi, V. K. and Singer, I. M. , title =. Math. Proc. Cambridge Philos. Soc. , fjournal =. 1975 , pages =

1975
[69]

M. A. Int. Math. Res. Not. , fjournal =. 2011 , number =

2011
[70]

1990 , pages =

Hein, Wolfgang , title =. 1990 , pages =

1990
[71]

2012 , publisher =

M. 2012 , publisher =

2012
[72]

Lubotzky, Alexander and Mozes, Shahar and Raghunathan, M. S. , title =. Inst. Hautes \'Etudes Sci. Publ. Math. , fjournal =. 2000 , pages =

2000
[73]

Fried, David , title =. Invent. Math. , fjournal =. 1986 , number =

1986
[74]

Fried, David , title =. Comm. Math. Phys. , fjournal =. 1995 , number =

1995
[75]

Ray, D. B. and Singer, I. M. , title =. Advances in Math. , fjournal =. 1971 , pages =

1971
[76]

, title =

Wallach, Nolan R. , title =. Bull. Amer. Math. Soc. , fjournal =. 1976 , number =

1976
[77]

Wotzke , title =

A. Wotzke , title =. 2008 , address =

2008
[78]

Pfaff, Jonathan , title =
[79]

Barbasch, Dan and Moscovici, Henri , title =. J. Funct. Anal. , fjournal =. 1983 , number =

1983
[80]

, title =

Chernoff, Paul R. , title =. J. Functional Analysis , volume =. 1973 , pages =

1973
[81]

, title =

Miatello, Roberto J. , title =. Trans. Amer. Math. Soc. , fjournal =. 1980 , number =

1980
[82]

, title =

Miatello, Roberto J. , title =. Manuscripta Math. , fjournal =. 1979 , number =

1979
[83]

2000 , pages =

Dirac operators in. 2000 , pages =

2000
[84]

1963 , pages =

Kobayashi, Shoshichi and Nomizu, Katsumi , title =. 1963 , pages =

1963
[85]

Blaine and Michelsohn, Marie-Louise , title =

Lawson, Jr., H. Blaine and Michelsohn, Marie-Louise , title =. 1989 , pages =

1989
[86]

Aluetta , title =

G. Aluetta , title =. 2001 , publisher =

2001
[87]

Hejhal , title =

Dennis A. Hejhal , title =. 1976 , address =

1976
[88]

Selberg , title =

A. Selberg , title =
[89]

M?ller and J

W. M?ller and J. Pfaff , title =. Intern. Math. Research Notices , year =
[90]

Müller, Werner and Pfaff, Jonathan , title =. Int. Math. Res. Not. ImrN , fjournal =. 2013 , number =

2013
[91]

Arthur , title =

J. Arthur , title =. Duke. Math. J. , volume =. 1978 , pages =

1978
[92]

W. M. Limit multiplicities for principal congruence subgroups of
[93]

Gel'fand, I. M. and Graev, M. I. and pyatetskii-Shapiro, I. I. , title =. 1969 , pages =

1969
[94]

, title =

Gilkey, Peter B. , title =. 1995 , pages =

1995
[95]

1994 , publisher =

Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem , author =. 1994 , publisher =

1994
[96]

2012 , month =

Müller, Werner , title =. 2012 , month =

2012
[97]

Müller, Werner , title =. J. Amer. Math. Soc. , fjournal =. 1993 , number =

1993
[98]

On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds , journal =

Marshall, Simon and M. On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds , journal =. 2013 , number =

2013
[99]

The analysis of linear partial differential operators

H\". The analysis of linear partial differential operators. 2003 , pages =

2003
[100]

Reed, Michael and Simon, Barry , title =
[101]

, title =

Knapp, Anthony W. , title =. 1986 , pages =

1986
[102]

Acta Math

Harish-Chandra , title =. Acta Math. , fjournal =. 1965 , pages =

1965
[103]

Acta Math

Harish-Chandra , title =. Acta Math. , fjournal =. 1966 , pages =

1966
[104]

Harish-Chandra , title =. Trans. Amer. Math. Soc. , fjournal =. 1965 , pages =

1965
[105]

Helgason , title =

S. Helgason , title =. 1984 , publisher =

1984
[106]

On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds , journal =

Matsushima, Yoz. On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds , journal =. 1963 , pages =

1963
[107]

Riemann surfaces and related topics:

Fay, John , title =. Riemann surfaces and related topics:
[108]

Braverman, Maxim and Kappeler, Thomas , title =. J. Differential Geom. , fjournal =. 2008 , number =

2008
[109]

Braverman, Maxim and Kappeler, Thomas , title =. C. R. Math. Acad. Sci. Paris , fjournal =. 2005 , number =

2005
[110]

Shubin, M. A. , title =. 1987 , pages =

1987
[111]

, title =

Bismut, Jean-Michel and Freed, Daniel S. , title =. Comm. Math. Phys. , fjournal =. 1986 , number =

1986
[112]

Markus, A. S. , title =. 1988 , pages =

1988
[113]

2005 , pages =

Simon, Barry , title =. 2005 , pages =

2005
[114]

1997 , pages =

Bump, Daniel , title =. 1997 , pages =

1997
[115]

, title =

Wallach, Nolan R. , title =. 1973 , pages =

1973
[116]

1996 , pages =

Kobayashi, Shoshichi and Nomizu, Katsumi , title =. 1996 , pages =

1996
[117]

Cheeger, Jeff and Yau, Shing Tung , title =. Comm. Pure Appl. Math. , fjournal =. 1981 , number =

1981
[118]

, title =

Onishchik, Arkady L. , title =. 2004 , pages =

2004
[119]

, title =

Moscovici, Henri and Stanton, Robert J. , title =. Invent. Math. , fjournal =. 1989 , number =

1989
[120]

, title =

Goodman, Roe and Wallach, Nolan R. , title =. 1998 , pages =

1998
[121]

, title =

Gilkey, Peter B. , title =. Algebraic and differential topology---global differential geometry , series =
[122]

1993 , pages =

Jorgenson, Jay and Lang, Serge , title =. 1993 , pages =

1993
[123]

, title =

Voros, A. , title =. Comm. Math. Phys. , fjournal =. 1987 , number =

1987
[124]

Ruelle, David , title =. Bull. Amer. Math. Soc. (N.S.) , fjournal =. 2004 , number =

2004
[125]

Ergodic Theory Dynam

Ruelle, David , title =. Ergodic Theory Dynam. Systems , fjournal =. 2008 , number =

2008
[126]

Ruelle, David , title =. Comm. Math. Phys. , fjournal =. 2007 , number =

2007
[127]

and Liverani, C

Giulietti, P. and Liverani, C. and Pollicott, M. , title =. Ann. of Math. (2) , fjournal =. 2013 , number =

2013
[128]

and Klingenberg, W

Gromoll, D. and Klingenberg, W. and Meyer, W. , title =. 1968 , pages =

1968
[129]

Tables of integral transforms

Erd. Tables of integral transforms. 1954 , pages =

1954
[130]

, title =

Grubb, Gerd and Seeley, Robert T. , title =. Invent. Math. , fjournal =. 1995 , number =

1995
[131]

Braverman, Maxim and Kappeler, Thomas , title =. Geom. Topol. , fjournal =. 2007 , pages =

2007
[132]

arXiv preprint arXiv:1602.00664 , year =

Analytic torsion, dynamical zeta functions and orbital integrals , author =. arXiv preprint arXiv:1602.00664 , year =

arXiv
[133]

Analytic torsion, dynamical zeta functions, and the

Shen, Shu , journal =. Analytic torsion, dynamical zeta functions, and the. doi:10.2140/apde.2018.11.1 , year =

2018 doi
[134]

arXiv preprint arXiv:1306.4203 , year =

Dynamical zeta functions for Anosov flows via microlocal analysis , author =. arXiv preprint arXiv:1306.4203 , year =

arXiv
[135]

Journal of Functional Analysis , volume =

Horizontal lift of the Brownian motion on the hyperbolic plane and the Selberg trace formula , author =. Journal of Functional Analysis , volume =
[136]

arXiv preprint arXiv:1503.00525 , year =

Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations , author =. arXiv preprint arXiv:1503.00525 , year =

arXiv
[137]

Inventiones mathematicae , volume =

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds , author =. Inventiones mathematicae , volume =. 2012 , publisher =

2012
[138]

arXiv preprint arXiv:1307.4912 , year =

Refined analytic torsion as analytic function on the representation variety and applications , author =. arXiv preprint arXiv:1307.4912 , year =

arXiv
[139]

2011 , series =

A Course in Complex Analysis: From Basic Results to Advanced Topics , author =. 2011 , series =

2011
[140]

Journal f

Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume , author =. Journal f
[141]

Analytic torsion versus

Pfaff, Jonathan , journal =. Analytic torsion versus. 2014 , publisher =

2014
[142]

Analytic torsion and

Park, Jinsung , journal =. Analytic torsion and. 2009 , publisher =

2009
[143]

Inventiones mathematicae , volume =

R-torsion and zeta functions for locally symmetric manifolds , author =. Inventiones mathematicae , volume =. 1991 , publisher =

1991
[144]

Gradshteyn, I. S. and Ryzhik, I. M. , TITLE =. 2007 , PAGES =

2007
[145]

An approach to the

Fischer, J\". An approach to the. 1987 , PAGES =. doi:10.1007/BFb0077696 , URL =

1987 doi
[146]

An asymptotic expansion of the double gamma function , JOURNAL =

Ferreira, Chelo and L\'. An asymptotic expansion of the double gamma function , JOURNAL =. 2001 , NUMBER =. doi:10.1006/jath.2001.3578 , URL =

2001
[147]

, TITLE =

Adamchik, Victor S. , TITLE =. Int. J. Math. Comput. Sci. , FJOURNAL =. 2014 , NUMBER =

2014
[148]

Barnes, E. W. , title =. Quart. J. , volume =. 1900 , zbMATH =

1900
[149]

, TITLE =

Gilkey, Peter B. , TITLE =. 1995 , PAGES =

1995
[150]

The twisted

B\'. The twisted. J. \'. 2023 , PAGES =. doi:10.5802/jep.247 , URL =

2023 doi
[151]

Fuchsian groups and

Fried, David , journal=. Fuchsian groups and. 1986 , publisher=

1986
[152]

2020 , publisher=

Fedosova, Ksenia and Pohl, Anke , journal=. 2020 , publisher=

2020
[153]

Hoffmann, Werner , TITLE =. Ann. Global Anal. Geom. , FJOURNAL =. 1994 , NUMBER =

1994
[154]

2022 , DOI =

Shen, Shu and Yu, Jianqing , URL =. 2022 , DOI =

2022
[155]

International Mathematics Research Notices , volume=

Complex valued analytic torsion and dynamical zeta function on locally symmetric spaces , author=. International Mathematics Research Notices , volume=. 2023 , publisher=

2023
[156]

Determinants of

Naud, Fr. Determinants of. Journal d'Analyse Math. 2023 , publisher=

2023
[157]

Determinants of

Sarnak, Peter , journal=. Determinants of. 1987 , publisher=

1987
[158]

Yamaguchi, Yoshikazu , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2022 , PAGES =. doi:10.1515/crelle-2021-0075 , URL =

2022 doi
[159]

The resolvent kernel on the discrete circle and twisted cosecant sums , JOURNAL =

Jorgenson, Jay and Karlsson, Anders and Smajlovi\'. The resolvent kernel on the discrete circle and twisted cosecant sums , JOURNAL =. 2024 , NUMBER =. doi:10.1016/j.jmaa.2024.128454 , URL =

2024
[160]

Indiana Univ

Yamaguchi, Yoshikazu , TITLE =. Indiana Univ. Math. J. , FJOURNAL =. 2017 , NUMBER =. doi:10.1512/iumj.2017.66.6012 , URL =

2017 doi
[161]

Duke Math

Jorgenson, Jay , TITLE =. Duke Math. J. , FJOURNAL =. 1991 , NUMBER =. doi:10.1215/S0012-7094-91-06222-8 , URL =

1991 doi
[162]

Ray, D. B. and Singer, I. M. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1973 , PAGES =. doi:10.2307/1970909 , URL =

1973 doi
[163]

Ast\'erisque , FJOURNAL =

Maillot, Vincent and R\"ossler, Damian , TITLE =. Ast\'erisque , FJOURNAL =. 2009 , PAGES =

2009
[164]

Yoshikawa, Ken-ichi , TITLE =. J. Differential Geom. , FJOURNAL =. 1999 , NUMBER =

1999
[165]

Yoshikawa, Ken-Ichi , TITLE =. Invent. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1007/s00222-003-0334-3 , URL =

2004 doi
[166]

and Kramer, J

Jorgenson, J. and Kramer, J. , TITLE =. Duke Math. J. , FJOURNAL =. 2001 , NUMBER =. doi:10.1215/S0012-7094-01-10635-2 , URL =

2001 doi
[167]

Riemann surfaces and related topics:

Fay, John , TITLE =. Riemann surfaces and related topics:. 1981 , ISBN =

1981
[168]

Spilioti, Polyxeni , TITLE =. Ann. Global Anal. Geom. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s10455-020-09715-8 , URL =

2020 doi
[169]

arXiv preprint arXiv:2402.02959 , year=

On the functional equation of twisted Ruelle zeta function and Fried's conjecture , author=. arXiv preprint arXiv:2402.02959 , year=

arXiv
[170]

Hiroshima Math

Deitmar, Anton , TITLE =. Hiroshima Math. J. , FJOURNAL =. 2019 , NUMBER =. doi:10.32917/hmj/1564106546 , URL =

2019
[171]

, TITLE =

Berthomieu, A. , TITLE =. J. Math. Phys. , FJOURNAL =. 2001 , NUMBER =. doi:10.1063/1.1390329 , URL =

2001 doi
[172]

and Kramer, J

Jorgenson, J. and Kramer, J. , TITLE =. Manuscripta Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1007/s002290050071 , URL =

1998 doi
[173]

and Salvati Manni, R

Kramer, J. and Salvati Manni, R. , TITLE =. Abh. Math. Semin. Univ. Hambg. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s12188-008-0001-0 , URL =

2008 doi
[174]

Duke Math

Tsuzuki, Masao , TITLE =. Duke Math. J. , FJOURNAL =. 1997 , NUMBER =. doi:10.1215/S0012-7094-97-08802-5 , URL =

1997 doi
[175]

The asymptotics of the

M\". The asymptotics of the. Metric and differential geometry , SERIES =. 2012 , ISBN =. doi:10.1007/978-3-0348-0257-4\ _ 11 , URL =

2012 doi
[176]

Menal-Ferrer, Pere and Porti, Joan , TITLE =. J. Topol. , FJOURNAL =. 2014 , NUMBER =. doi:10.1112/jtopol/jtt024 , URL =

2014 doi
[177]

, title =

Friedman, Joshua S. , title =. Commun. Math. Phys. , issn =. 2007 , doi =

2007
[178]

, title =

Hejhal, Dennis A. , title =. 1983 , publisher =. doi:10.1007/bfb0061302 , keywords =

1983 doi
[179]

Gong, Donggeng , title =. Manuscr. Math. , issn =. 1995 , doi =

1995
[180]

On the error term of the prime geodesic theorem , JOURNAL =

Jorgenson, Jay and Kramer, J\". On the error term of the prime geodesic theorem , JOURNAL =. 2002 , NUMBER =. doi:10.1515/form.2002.040 , URL =

2002 doi
[181]

and Kramer, J

Jorgenson, J. and Kramer, J. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2001 , PAGES =. doi:10.1515/crll.2001.092 , URL =

2001 doi
[182]

, TITLE =

Huxley, M. , TITLE =. Automorphic forms and analytic number theory (. 1990 , ISBN =

1990
[183]

Effective bounds for

Avdispahi\'. Effective bounds for. Math. Comp. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/mcom/3631 , URL =

2021 doi
[184]

Friedman, J. S. and Jorgenson, J. and Kramer, J. , TITLE =. Math. Comp. , FJOURNAL =. 2011 , NUMBER =. doi:10.1090/S0025-5718-2010-02430-5 , URL =

2011 doi
[185]

2010 , publisher =

Buser, Peter , title =. 2010 , publisher =. doi:10.1007/978-3-7643-8878-2 , keywords =

2010 doi
[186]

Wu, Yunhui and Xue, Yuhao , title =. J. Eur. Math. Soc. (2025) published online , issn =. 2025 , doi =

2025
[187]

1984 , publisher =

Chavel, Isaac , title =. 1984 , publisher =

1984
[188]

Duke Math

Le Masson, Etienne and Sahlsten, Tuomas , title =. Duke Math. J. , issn =. 2017 , doi =

2017
[189]

Monk, Laura , title =. Anal. PDE , issn =. 2022 , doi =

2022
[190]

Spectral

Laura Monk and Fr. Spectral. 2026 , howpublished =

2026
[191]

Mirzakhani, Maryam , title =. J. Differ. Geom. , issn =. 2013 , doi =

2013
[192]

Mirzakhani, Maryam , title =. Invent. Math. , issn =. 2007 , doi =

2007
[193]

Wu, Yunhui and Xue, Yuhao , title =. Geom. Funct. Anal. , issn =. 2022 , doi =

2022
[194]

Lipnowski, Michael and Wright, Alex , title =. Ann. Probab. , issn =. 2024 , doi =

2024
[195]

2026 , howpublished =

Nalini Anantharaman and Laura Monk , title =. 2026 , howpublished =

2026
[196]

2025 , howpublished =

Nalini Anantharaman and Laura Monk , title =. 2025 , howpublished =

2025
[197]

A random cover of a compact hyperbolic surface has relative spectral gap

Magee, Michael and Naud, Fr. A random cover of a compact hyperbolic surface has relative spectral gap. Geom. Funct. Anal. , issn =. 2022 , doi =

2022
[198]

Brooks, Robert and Makover, Eran , title =. J. Differ. Geom. , issn =. 2004 , doi =

2004
[199]

and Sarnak, P

Buser, P. and Sarnak, P. , title =. Invent. Math. , issn =. 1994 , doi =

1994
[200]

, title =

Cheeger, J. , title =. 1970 , howpublished =

1970
[201]

Buser, Peter , title =. Ann. Sci. 1982 , doi =

1982
[202]

2024 , howpublished =

Giacomo Gavelli and Claudius Kamp , title =. 2024 , howpublished =

2024
[203]

Fortier Bourque, Maxime and Petri, Bram , title =. Am. J. Math. , issn =. 2022 , doi =

2022
[204]

Duke Math

Otal, Jean-Pierre and Rosas, Eulalio , title =. Duke Math. J. , issn =. 2009 , language =. doi:10.1215/00127094-2009-048 , keywords =

2009 doi
[205]

He, Yuxin and Wu, Yunhui , title =. J. Lond. Math. Soc., II. Ser. , issn =
[206]

Bergeron, Nicolas and Venkatesh, Akshay , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2013 , NUMBER =. doi:10.1017/S1474748012000667 , URL =

2013 doi
[207]

and Taimanov, Iskander A

Kordyukov, Yuri A. and Taimanov, Iskander A. , title =. Regul. Chaotic Dyn. , issn =. 2022 , doi =

2022
[208]

and Taimanov, Iskander A

Kordyukov, Yuri A. and Taimanov, Iskander A. , title =. Russ. Math. Surv. , issn =. 2019 , doi =

2019
[209]

Fay, J. D. , title =. J. Reine Angew. Math. , issn =. 1977 , doi =

1977
[210]

Charles, Laurent and Lefeuvre, Thibault , title =. J. 2026 , doi =

2026
[211]

Taimanov, I. A. , title =. Theor. Math. Phys. , issn =. 2024 , doi =

2024
[212]

Comtet, Alain , title =. Ann. Phys. , issn =. 1986 , doi =

1986
[213]

Construction of

Kara, Yasemin and Kumari, Moni and Marzec, Jolanta and Maurischat, Kathrin and Mocanu, Andreea and Smajlovi. Construction of. Women in numbers Europe III. Research directions in number theory. Selected papers based on the presentations at the 3rd conference, WINE 3, La Hublais...

2019 doi

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

[1] [1]

Turaev, V. G. , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 1986 , NUMBER =

1986

[2] [2]

Journal of the Institute of Mathematics of Jussieu , author=

Twisted. Journal of the Institute of Mathematics of Jussieu , author=. 2025 , pages=

2025

[3] [3]

Functional Analysis and Its Applications , volume=

Determinants of Cauchy-Riemann operators over a Riemann surface , author=. Functional Analysis and Its Applications , volume=. 1985 , publisher=

1985

[4] [4]

An extension of a theorem by Cheeger and M

Bismut, Jean-Michel and Zhang, Weiping , journal=. An extension of a theorem by Cheeger and M

[5] [5]

Geometry & Topology , volume=

Reidemeister torsion, complex volume and the Zograf infinite product for hyperbolic 3--manifolds , author=. Geometry & Topology , volume=. 2019 , publisher=

2019

[6] [6]

Inventiones mathematicae , volume=

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds , author=. Inventiones mathematicae , volume=. 2022 , publisher=

2022

[7] [7]

arXiv preprint arXiv:2303.00312 , year=

A Ruelle dynamical zeta function for equivariant flows , author=. arXiv preprint arXiv:2303.00312 , year=

arXiv

[8] [8]

The twisted

B. The twisted. arXiv preprint arXiv:2110.06683 , year=

arXiv