Spectral determinants of Laplacians on spheres

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Numbers

$d$

convention

$D_d\text{ or }L_d$

$D_d$:

39.47841760435743447533796399950460454125479762896316250565339750488017928967682097200709361487420893

$L_d$:

3.675754132818690967121318945622470559445589894551133651268606161931062783709041590778973194543816790

$D_d$:

3.195311486059186083954018930320625869027025537218383410459453552355138055220260136541487597410598484

$L_d$:

1.161684574801803716855678640971122571056145521340807134666089225429438798666308691038100401330035022

$D_d$:

3.338851214151637978641073442361581066827638921418583994784327754257705771814723420836902357386573631

$L_d$:

1.205626799966186715703930412295368264941295780005259444019566851190213405774699877139212440037238700

$D_d$:

1.736943483345117045218706484107328288727257562783357365404074864447438508652895789852662929429579563

$L_d$:

0.5521269497953281344354619746559145187328907971349279394472709784288480706712760686935962773799220630

$D_d$:

1.762919348160542197769112247656987373525615685549134467477465884088514659928021043170117532508763448

$L_d$:

0.5669711554382509653370433182345105700677569165837399269597178498193843266570763661172151916608754244

$D_d$:

1.290018366027266297051264330431495830757884049324567992644432599376220300780519067233745900131075388

$L_d$:

0.2546564555026713334537814714378304036356329955621741809601869507908325152156131560193201665354494393

$D_d$:

1.222521243741051798580285546613412059282832656608533157852240761731551116252190768511571643543843445

$L_d$:

0.2009153195092005313671752538232644823103729233242375018102284417546963249821077585606706813448139037

$D_d$:

1.050409114156376719441761195267729890047486147797145919662748458556665692080673826640793755647776124

$L_d$:

0.04917972081227182205829640227742091843407034364894111039130138905323558606445435820638864599173862917

$D_d$:

0.9467334221847900636684050677635285693801266468093952263403606529619547215600289645849774869799206801

$L_d$:

-0.05473772258877507042606709280782940534969410181824304090728365084837904809699410346299349060928330897

$D_d$:

0.8961828605681046317716580461174625600305612889264271896291549692524329293702628739548211567022235446

$L_d$:

-0.1096108013743362174616142966023982225844041424225712613964170930271188257712523573877834821009474310

$D_d$:

0.7780483648571937970816548672265559050285317701003298226457651654161308289551019660021614796152839316

$L_d$:

-0.2509665911097348069306166614379815554524909497122545702189146247458269560478639863392089329773701560

$D_d$:

0.7869039135647151410517427669687195098699574958017690586310546317962197939278418801259812527206671209

$L_d$:

-0.2396491300583256639079333379866968663412497495802051874859446323872315589634331378453165044926199903

$D_d$:

0.6635456412554321342343814345854801243050403557897350370721523223354367228096232540025646171739917565

$L_d$:

-0.4101576389465675069863080104315532664254895548376359214342597585785548280839153854937079214554798474

$D_d$:

0.7046548726983077273396470668842922877078604841208326779468479199547437598151776816004846484380394592

$L_d$:

-0.3500471383067112218206201100269335398334451685735282344924553723793408362379678998247896487735267074

$D_d$:

0.5803750864624722845692971576602470017915947264346365847350023894712364297959165581248202857301247541

$L_d$:

-0.5440806836651555685925915007868069530040812890697187173360803433919152866411553801836091995744143649

$D_d$:

0.6401080468695482034108900862465628799057371366409760577624327081392960768555661091965747207502690090

$L_d$:

-0.4461182936437888670078446901438182663566279309577078529473244169263530072799737148431416737709949589

$D_d$:

0.5170199343083330469374813463401077991474108751117166058297717260054855118836789084550852570986482612

$L_d$:

-0.6596738475606850580357341973100950205815305549216960249557074353633101356293982856177649573431972653

$D_d$:

0.5878700716168702112163332445108552806398959455050943848841430037745321198674719299601899749764241270

$L_d$:

-0.5312493221382104326335200853727827925218643933752426702082616447114821934698581245827253329813996814

$D_d$:

0.4670281557074978548724853588906024846732098731089579413535973564593172426553649984155737588830040241

$L_d$:

-0.7613657325364307459445768201896824488696654871639468641283522235096508119732706811788601984011218194

$D_d$:

0.5445809447748113338885678887533313943385324702616741670941521688236528647263512269708291547860468913

$L_d$:

-0.6077386887508543522150025855747422667524154202557010771525578982273659100163490654451995500049760936

Definition

The zeta-regularized determinant of the scalar Laplacian on the unit round sphere $S^d$, with its zero mode omitted [1]. Each dimension is given in two conventions: $D_d=\det{}^{\prime}\Delta_d$ and $L_d=\log D_d$.

Parameters

$d$

— dimension of the sphere ($d\geq1$)

convention

— determinant or natural logarithm

Formulas

(1)

For $d\geq2$, $\displaystyle Z_d(s)=\sum_{\ell=1}^{\infty}m_{d,\ell}[\ell(\ell+d-1)]^{-s}$, where $\displaystyle m_{d,\ell}=\frac{(2\ell+d-1)(\ell+d-2)!}{\ell!(d-1)!}$ and $\ell$ is the spherical-harmonic degree. For $d=1$, $Z_1(s)=2\zeta(2s)$, where $\zeta$ is the Riemann zeta function. [1], Section 2.1.

(2)

$D_d=e^{L_d}$ and $L_d=\log D_d=-Z_d^{\prime}(0)$.

(3)

$D_1=4\pi^2$ and $L_1=2\log(2\pi)$.

(4)

$D_2=e^{1/6}A^4$ and $L_2=1/6+4\log A$, where $A$ is the Glaisher–Kinkelin constant. [1], Corollary 2.9.

(5)

$\displaystyle D_3=\pi\exp\!\left(\frac{\zeta(3)}{2\pi^2}\right)$ and $\displaystyle L_3=\log\pi+\frac{\zeta(3)}{2\pi^2}$. [1], Corollary 2.9.

Comments

(6)

The sphere is $S^d=\{x\in\mathbb{R}^{d+1}:|x|=1\}$ with its induced round metric. The nonnegative operator $\Delta_d=-\operatorname{div}\operatorname{grad}$ acts on scalar functions, with no additive shift or volume normalization. The dimension convention is the same as in the sphere-volume table.

(7)

The spectral zeta function $Z_d(s)$ is continued meromorphically from $\Re s>d/2$ and is regular at $s=0$. Positive eigenvalues use real logarithms. The prime on $\det{}^{\prime}$ omits the constant zero mode; the prime on $Z_d$ denotes differentiation with respect to $s$.

(8)

The determinant $D_d>0$ and its natural logarithm $L_d$ are one number in two invertible conventions. They are stored for the same dimensions.

(9)

The finite formulas in [1], Theorems 2.7 and 2.8, express $L_d$ through Riemann zeta derivatives at nonpositive integers and elementary terms.

Programs

(P1)

Sage

# Download the attached generate.py into the current directory.
from generate import logdet
for d in range(1, 21):
    L = logdet(d, working_digits=150)
    print(d, L.exp(), L)

References

[1]

José Cunha and Pedro Freitas, Recurrence formulae for spectral determinants, 2024, Section 2 and Appendix B. (arXiv)

Links

[2]

Cunha and Freitas: full text and numerical comparison table

Similar tables

Volume of the unit sphere — holds volumes of the same unit round spheres, with the same dimension convention

Dirichlet spectral determinants of homogeneous oscillators — applies spectral zeta regularization to half-line Schrödinger operators with a Dirichlet boundary condition

Neumann spectral determinants of homogeneous oscillators — applies spectral zeta regularization to half-line Schrödinger operators with a Neumann boundary condition

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every integer dimension $1\leq d\leq20$ in both determinant and natural-logarithm conventions; this noncanonical practical prefix covers the circle, low-dimensional closed forms and ten dimensions of each parity, providing breadth across both parity formulas without extending to a long high-dimensional listing)

How they were obtained:

The finite formulas of [1], Theorems 2.7 and 2.8, were evaluated with exact integer and rational coefficients and Arb ball arithmetic for zeta derivatives, logarithms and exponentials. All 40 values carry at least 505 bits of relative accuracy at 150 working decimal digits and overlap recomputations at 200 digits; 100 significant digits are stored.

A separate continuation of the multiplicity-weighted spectral sum, with an explicit tail estimate, agrees in all 20 dimensions to at least 160 decimal places. That independent check uses mpmath with extra precision for small Hurwitz zeta values and is corroboration, not the basis of the proven enclosure. Nonzero controls test dimensions 1, 2 and 3, every negative-even zeta derivative is checked through the functional equation, and all determinants agree with the rounded source Table 2. The attached generator, checks and numerical evidence reproduce these results.