Spectral determinants of the Laplacian on real projective spaces

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Numbers

$d$

convention

$D_d\text{ or }L_d$

$D_d$:

9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232

comment: $D_1=\pi^2$.

$L_d$:

2.289459771698800348286854702706117423294589625830623143027246142944275539769652159567246540550979415

comment: $L_1=2\log\pi$.

$D_d$:

2.240353442041525736384909576289485062286128606841744714702197445650291135167456934833947796246473099

$L_d$:

0.8066336400456292907909369354330023573139700999479317260295161437040914732907200276079301723045530212

$D_d$:

2.004050458417333083781669314716021886697441120867330400882094627120002124252284040368585537045496100

$L_d$:

0.6951703617566010309682074736641203567477985469148478074167181808510466914746265536001166223260669879

$D_d$:

1.312925012773452498079424837950309503433576021467837141542963324038584252439466081655821254051149637

$L_d$:

0.2722574823214659970758470537612606336221025361558437332089646969049460511057366691157133741718127049

$D_d$:

1.226324767387671873211881595676757782358511989155926149103412399481523969547820135132271799907023864

$L_d$:

0.2040217024241567091909621201976406736696681095849377487039527244198537603494593404246296858712959854

$D_d$:

0.9853902801411974028421197679266859744895737904378955061863754318023804286097645549916656565615329407

$L_d$:

-0.01471749279224656792335568021414392516811418617320029458601131685000511374222687187041850606027962850

$D_d$:

0.9093136996128456617980119839958127653054609782553909044040492806425597827318874159851846149052554498

$L_d$:

-0.09506514025483739680890530391009428515221403727419613654824706943133459989638842044090953073374235420

$D_d$:

0.8043769179373029692858918090292417447353530174363870235089224389412588345151913728434722141551946265

$L_d$:

-0.2176873162525491436797055654537276094195669413497451976942936515017610401005653821064991706366261763

$D_d$:

0.7325502848617156598391933979224193944352110783360670592837268931394173534837089295183376346388433772

$L_d$:

-0.3112232921717199654333670116663055156562402799145098939219888686551750884863367591649629837407372932

$D_d$:

0.6865186724655433430176133136180018317260805714076113737466305596092352905712505329831981258091565610

$L_d$:

-0.3761218546312024048861602222854568388981594221872191781218697293092689752941226503411738110369675353

$D_d$:

0.6182213888372435506900607088845110872503155582746300794543068687632795654437930189672019622178010479

$L_d$:

-0.4809086513105842818442992789844195166245622378784436150766806397476649294931351190965660083742407130

$D_d$:

0.6025577230142032069622642883801304171335259868423480896936395866220132108155892911751338198077049877

$L_d$:

-0.5065718123737102580010260132503578751790074499266845224606288457962278657429385458528419403117347040

$D_d$:

0.5375178289620966741514838835563268030899579297824644213877458132177142294936930341403974901018800668

$L_d$:

-0.6207933493683965247237718646265587676232459684306682852085393181275159426659407143099086715669268037

$D_d$:

0.5391892045770326418554860025568188598644296291531152576229520498670461537163627573880437329447247466

$L_d$:

-0.6176887407694948572660795742423246982845345842618878146939934993256911275771520386420414224443112595

$D_d$:

0.4771574115925355031993313685202747117841437316370279378113806643351300379970146070048295854047563482

$L_d$:

-0.7399088391942723417040637449558431334074174153682087057803793294100102770301244706650254294845768011

$D_d$:

0.4893869214411606634757739627163906211326364567018052780136915655158061345453826009712720696695144724

$L_d$:

-0.7146018519899039842520085023676804533664249847558121888444496402436878785208006771373361519952422907

$D_d$:

0.4301144107908169280307533361661010714960870225034133697413646725338131594168017235909898685563239563

$L_d$:

-0.8437040340787503415173139645183573405826063524685599829955640653294324599701927674189887109781742654

$D_d$:

0.4490524518115977612885905833417110081625800932187691005284917291941009459309655422082164123144900818

$L_d$:

-0.8006155788832067196200959305662654992057889855679047411575227128953489582081950245224597331656394929

$D_d$:

0.3923023205323878556030533698230096399187238960117325217189618753045058421585312707304510103831652540

$L_d$:

-0.9357225105855833262591619126748270353284174934000451061228403712368581031315954771558670390407778353

$D_d$:

0.4156177744028258362737485267831917234676954383886315507850366660063977210569258631966261182514476567

$L_d$:

-0.8779892526185941469999085077321031028703225581903123173437164805333762172069365846575001471592285302

Definition

$D_d$ is the zeta-regularized determinant of the nonnegative scalar Laplacian on $\mathbb{RP}^d=S^d/(x\sim-x)$ with the quotient metric of the radius-one round sphere and its constant zero mode omitted [1]. The entries $D_d=\det{}^{\prime}\Delta_d$ and $L_d=\log D_d$ give one number in two invertible conventions.

Parameters

$d$

— dimension of real projective space ($d\geq1$)

convention

— determinant or natural logarithm

Formulas

(1)

$\displaystyle Z_d(s)=\sum_{k=1}^{\infty}m_{d,k}[2k(2k+d-1)]^{-s}$, where $\displaystyle m_{d,k}=\binom{d+2k}{2k}-\binom{d+2k-2}{2k-2}$ and $k$ is half the spherical-harmonic degree. This formula includes $d=1$. [1], Section 3.2; [2], Section 3.

(2)

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

(3)

$Z_1(s)=2^{1-2s}\zeta(2s)$, $D_1=\pi^2$ and $L_1=2\log\pi$, where $\zeta$ is the Riemann zeta function.

Comments

(4)

The operator is $\Delta_d=-\operatorname{div}\operatorname{grad}$ on ordinary scalar functions, equivalently the trivial line bundle. The metric is not rescaled to unit volume, and the operator has no mass or curvature shift. Antipodally invariant spherical harmonics have even degree, so this spectrum selects the even degrees of the sphere spectrum. [2], Section 3.

(5)

The spectral zeta function $Z_d(s)$ is continued meromorphically from $\Re s>d/2$ and is regular at zero. All eigenvalues in the sum are positive, fixing the real logarithm. The prime on the determinant omits the constant zero mode; the prime on $Z_d$ differentiates with respect to $s$.

(6)

The determinant $D_d>0$ and its natural logarithm $L_d$ are one number in two invertible conventions, stored over the same dimension range.

(7)

The space $\mathbb{RP}^1$ is a circle of circumference $\pi$. Its positive eigenvalues are $4k^2$, each with multiplicity two.

(8)

The finite expressions in [3], Theorem 3.3, give $L_d$ in terms of Riemann zeta derivatives at nonpositive integers and elementary terms. The circle is evaluated directly from its spectrum.

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 3.2, Theorem 3.3 and Appendices A and B. (arXiv)

[2]

Mauro Spreafico, Zeta functions and regularized determinants on projective spaces, 2001, Sections 2 and 3. (arXiv)

Links

[3]

Cunha and Freitas: formulas for real projective spaces

Similar tables

Spectral determinants of Laplacians on spheres — holds the scalar determinants on the radius-one round spheres before the antipodal quotient

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every dimension $1\leq d\leq20$ in determinant and natural-logarithm conventions for the radius-one quotient metric; this noncanonical practical prefix covers the circle, both parity formulas and five dimensions in each residue class modulo four, balancing geometric breadth against cancellation and the diminishing benefit of a longer high-dimensional listing)

How they were obtained:

The finite formulas of [1], Theorem 3.3, are evaluated with exact integer and rational coefficients and Arb ball arithmetic for zeta derivatives, logarithms and exponentials. All 40 values retain at least 503 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 defining multiplicity-weighted spectral sum agrees in all 20 dimensions to absolute error below $10^{-160}$ in the logarithm. Its omitted tail is bounded by $10^{-170}$ and it uses 190-digit mpmath arithmetic; this independent floating computation is corroboration, not the basis of the proven enclosure. The circle supplies a nonzero control, 1000 exact comparisons check the multiplicity polynomial against the binomial formula, and negative-even zeta derivatives are checked with the functional equation. Every determinant agrees with the rounded values in Appendix B of [1]. The attached generator, checks, continuation derivation and numerical evidence reproduce the calculation.