Neumann spectral determinants of regular polygons

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Numbers

$n$

convention

$D_n^N\text{ or }L_n^N$

$D_n^N$:

3.079442672013086778322018392320670368889792186044913220457624190067973131775759772891580258233990917

$L_n^N$:

1.124748629966838191330519136748164332501965014519228989431594805199304805923255644266267253342366265

$D_n^N$:

3.654316552624467285628970373604888358807855665573876133678786125063632037488036455853155933438320556

$L_n^N$:

1.295909086010958963534580078527643972284281623376695939515619730796599962724197670885685522456519945

$D_n^N$:

3.934418400393761555283407536299291558273452635642646094000438733872652551568761541048216546460978819

$L_n^N$:

1.369763069266781263411241299202357719387637452340263363725224190991083680517138650333432354151057094

$D_n^N$:

4.072626872583775047165893130487611583717758878203719733015394368607699222103978014965673184478714751

$L_n^N$:

1.404288214497759710572375928708608632224802585286781034549692562962598078198876459299587350466403175

$D_n^N$:

4.143340233580530652355728536670091119240628491756674448203223577922329804558047176126747667153765260

$L_n^N$:

1.421502282238609378949444160623898948094207147911312589692895244951592447159538628174726741230754625

$D_n^N$:

4.179747669222731784728453195425434266372990447465046429810421746776928978669627199987180403127091445

$L_n^N$:

1.430250878499894211892250703922562395161571138606487186563271387129776994850768297770626919530117513

$D_n^N$:

4.197693767143778453278181212860730790059537714495093412317509943526091339648098581917163419963005304

$L_n^N$:

1.434535271416021499917139408632637492436944941202307218979602047870547458756289511504594407255111320

$D_n^N$:

4.205273611715434167588381985608970512174584903297743264352295309544935259189785428857604233497908819

$L_n^N$:

1.436339359492476541268934607694689457297586463012863502022518393393445410350904862323343028896242936

$D_n^N$:

4.206863500720502733626523986965242124484029076903697672993997526803917695264278791913449310088047338

$L_n^N$:

1.436717358329810962049023557118968280683319753743850722348430862583051897918030442860729018170615610

$D_n^N$:

4.204939411524706318324792154863472060106794316047564035835531165964625395010135661557985543232058854

$L_n^N$:

1.436259884646838207574429745959819956064511584758978394397644394949481164389897315206376060053198164

$D_n^N$:

4.200953621866675648321160179930550574333772843235893815665758667921047957631686674681356628678681066

$L_n^N$:

1.435311552342129910992873130427476751155886248025332060207452523780450157659594091500865264904731569

$D_n^N$:

4.195782183375745459242500043020945486621318797479254571903741290354304955975833010345732622563599087

$L_n^N$:

1.434079778647029900426279380335915752788257147842409143626030185953797268031159246716111248915178793

$D_n^N$:

4.189964732006923311406803531686754698740507799388009106554402251312215036612490723449461890759240238

$L_n^N$:

1.432692316716503145374140290234629726574392596250665761167759390966338158303167171858520394248458157

$D_n^N$:

4.183838441963860012988250102243293012853143238395177281185334427276541485711547078086248028928282826

$L_n^N$:

1.431229112719752625985660893159898636935478361096721427947335337292678345689257401604428490042160627

$D_n^N$:

4.177615541871568481497457589340140948119930578917685879647375587455847930199159784903534702875075220

$L_n^N$:

1.429740639242950069388321873986112334742796403996376599212877200509260033358143819253475174758407101

$D_n^N$:

4.171429550094395127363904363781146577815135200457482772715070379203396494625510529758344189228872074

$L_n^N$:

1.428258794830565302268686326737056380245792203388435887152120895088617310758367397917264742519521711

$D_n^N$:

4.165363578059086385880020847239446537008700409334538841098868313795999529830493941524122933050358406

$L_n^N$:

1.426803565460578310373852021111493795053283312479696260712782858340137547052016417248752385895984739

$D_n^N$:

4.159468047082304840620350597222163634053951087849876346717278392977261851200226467175247221461800762

$L_n^N$:

1.425387192799161886888115784307198812919207024607713228630200574970419399194984767292500024843679875

$D_n^N$:

4.153771992173089190955107811911597625102492418091382753005924594153325356700413198933713679867605789

$L_n^N$:

1.424016835207249646286649181290151111628722153525892731330330683837644986902811543815259987424728915

$D_n^N$:

4.148290392100861632525663108327986586566212705118024722415272487931770251953109668332222571298364133

$L_n^N$:

1.422696295655730470291868920130582152839403226087445960165711062350049374717711067626231155609040325

$D_n^N$:

4.143028987711532956466258180606090755523178312548990093736349492626690309881211025373218625970558641

$L_n^N$:

1.421427159863336910111686315380493086425642219095068714707971203825499199103206404900633351505712615

$D_n^N$:

4.137987484565414140953067100393656370242524937227613331833278458582061768698694423364897647408372157

$L_n^N$:

1.420209554805827739410889517616281156059519704592070747774009137944912284313305832079828299230551300

$D_n^N$:

4.133161700211491816143621773456386079185941394432090253769430118585510456479514324608208125691656042

$L_n^N$:

1.419042658938349410082756638272475452757350620848929138085957320186159980034114301485158312752029228

$D_n^N$:

4.128545012791682953298054454234411092580026334306435376642641294148507384858509755159473921116168032

$L_n^N$:

1.417925047756852519981789884238925106076918444219149805564605352601177850939774322625575696175981487

$D_n^N$:

4.124129341715851853671192659019302723815347926783646073235643878308815631044989719990640048738533507

$L_n^N$:

1.416854928833571665449850181008274293440323629890598281833265197524790311723494230541949198521985059

$D_n^N$:

4.119905811830572872777330208347139341147642149743328799534546625143402765701948086857885894953633287

$L_n^N$:

1.415830301893942032179902031960900277850348012404268556644616783361278887532999144626340198691184650

$D_n^N$:

4.115865201736951830776716392512020371073568851273953579869573033446522833180836567604954602315687916

$L_n^N$:

1.414849067614561712616927855124614345820364780204292736112932858021772367580035189350654995387075423

$D_n^N$:

4.111998243935943966646091984408428714506464739960144226659556691700958616108061416517273289415552671

$L_n^N$:

1.413909101094393408967819016885025292560658880300789676819348695116345944559105720187940700669172056

Definition

For the circumradius-one regular polygon $P_n$, $D_n^N$ is the zeta-regularized Neumann Laplacian determinant with its constant zero mode omitted [3]. The determinant $D_n^N$ and its natural logarithm $L_n^N$ are one number in two invertible conventions.

Parameters

$n$

— number of sides ($n\geq3$)

convention

— determinant or natural logarithm

Formulas

(1)

$L_n^N=-(Z_n^N)^{\prime}(0),\qquad D_n^N=\det{}^{\prime}\Delta_N=\exp(L_n^N),\qquad L_n^N=\log D_n^N$ [3].

(2)

$\displaystyle \frac{D_n^N}{D_n^D}=A_n\alpha_n^{-n/2},\qquad L_n^N=L_n^D+\log A_n-\frac n2\log\alpha_n$, where $D_n^D$ and $L_n^D$ are the Dirichlet determinant and its natural logarithm [3], Theorem 1.1.

(3)

$\displaystyle c_n=Z_n^D(0)=\frac{n-1}{6(n-2)},\qquad Z_n^N(0)=c_n-1,\qquad R_n=\frac{\Gamma(1+1/n)\Gamma(1-2/n)}{\Gamma(1-1/n)}$, where $\Gamma$ is the Gamma function [1].

(4)

$\displaystyle L_n^N=2c_n\log R_n+\frac{\log n}{3(n-2)}-nF(\alpha_n)+\log A_n-\frac n2\log\alpha_n$ [1], Proposition 6, and [3], Theorem 1.1.

(5)

$\displaystyle F(a)=\zeta_B^{\prime}(0;a,1,1)-a\zeta_R^{\prime}(-1)+\frac{a-a^{-1}}{12}\log2-\frac{a-1}{4}\log(2\pi)$ for $a>0$, where $\zeta_B(s;a,1,1)=\sum_{j,k\geq0}(aj+k+1)^{-s}$ initially for $\Re s>2$, continued analytically to zero. The prime differentiates in $s$; $\zeta_R$ is the Riemann zeta function and $\zeta_R^{\prime}(-1)$ is its derivative at $-1$ [2], equation (3.15).

(6)

$\displaystyle L^N(tP_n)=L_n^N-2(c_n-1)\log t,\qquad D^N(tP_n)=t^{-2(c_n-1)}D_n^N\quad(t>0)$. Side length one uses $t=[2\sin(\pi/n)]^{-1}$, and area one uses $t=A_n^{-1/2}$. The term $-1$ in $Z_n^N(0)$ accounts for the omitted constant mode [4].

(7)

$\displaystyle D_3^N=\frac{3^{25/12}\Gamma(1/3)}{4\pi^{2/3}},\qquad D_4^N=2^{9/4}\eta(i)=\frac{2^{5/4}\Gamma(1/4)}{\pi^{3/4}}$, where $\eta(i)$ is the Dedekind eta value for the square lattice. These follow by adding the Neumann axis spectra to [4], Corollaries 2.2 and 3.3.

Comments

(8)

The domain is $P_n=\operatorname{int}\operatorname{conv}\{e^{2\pi i j/n}:0\leq j<n\}$, centred at zero with one vertex at one. Its interior angle is $\pi\alpha_n$, where $\alpha_n=1-2/n$, and its area is $A_n=(n/2)\sin(2\pi/n)$, the circumradius-one convention in the regular-polygon area table.

(9)

The operator is the nonnegative scalar Euclidean Laplacian $-\partial_x^2-\partial_y^2$. Homogeneous Neumann conditions hold on every side, with the usual selfadjoint realization. Write $Z_n^N(s)=\sum_{\lambda>0}\lambda^{-s}$ for $\Re s>1$, counting eigenvalues with multiplicity, and continue it analytically to zero. Exactly one constant zero mode is omitted. There is no mass term or area division; all logarithms are natural and real. The prime in $\det{}^{\prime}\Delta_N$ denotes zero-mode omission, while a prime on $Z_n^N$ denotes differentiation in $s$ [3].

(10)

The Dirichlet determinant table uses the same polygons and stores $D_n^D$ and $L_n^D=\log D_n^D$. The ratio in (2) follows from [3], Theorem 1.1: for a topological disk the squared torsion-norm quotient is its area. These polygons have flat smooth interiors, straight sides and positive corner angles, as required there. The source is a preprint dated 26 October 2026.

(11)

The Neumann spectrum table includes the constant zero mode and gives individual eigenvalues for the square and equilateral triangle of side length one. Here those shapes have side lengths $\sqrt2$ and $\sqrt3$. Their positive spectra equal the corresponding Dirichlet spectra together with two positive axis sequences; their determinant ratios are therefore $8$ and $27/4$, respectively [4].

(12)

The auxiliary scale $R_n$ in (3) is the reciprocal of the interior conformal radius $r(P_n,0)$ in its Euclidean convention. It is not the physical circumradius, which is fixed at one.

Programs

(P1)

Sage

# Download the attached generate.py into the current directory.
from generate import PolygonNeumannDeterminants
g = PolygonNeumannDeterminants()
print(g.value({'n': '31', 'convention': 'det'}, 100))

References

[1]

E. Aurell and P. Salomonson, On Functional Determinants of Laplacians in Polygons and Simplices, Communications in Mathematical Physics 165 (1994), 233–260. Proposition 6 and equations (18), (20), (23)–(25). (arXiv) (doi)

[2]

V. Kalvin, Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities, version 2 (2020). Equation (3.15) and Appendix A, Lemma A.1. (arXiv) (doi)

[3]

V. Kalvin, Determinants of Dirichlet and Neumann Laplacians on Constant Curvature Surfaces with Geodesic Polygonal Boundary, preprint, version 1, 26 October 2026. Section 1, Theorem 1.1, and Section 7. (arXiv)

[4]

G. Mårdby and J. Rowlett, Spectral Invariants of Integrable Polygons, Journal of Fourier Analysis and Applications 31 (2025), article 69. (doi)

Similar tables

Dirichlet spectral determinants of regular polygons — holds the companion determinant for the same domains; the Neumann/Dirichlet ratio is $A_n\alpha_n^{-n/2}$

Neumann eigenvalues of classical planar domains — holds individual eigenvalues and includes the zero mode; its square and triangle use side length one

Area of the regular n-gon — its circumradius-one branch gives the area factor in the determinant ratio

Interior conformal radii of regular polygons — its Euclidean values are the reciprocals of the scale $R_n$ in the finite determinant formula

Logarithmic capacity of regular polygons — holds the exterior conformal invariant of the same domains

Neumann spectral determinants on hemispheres — uses the same omitted-zero-mode convention on curved domains

Dirichlet spectral determinants on hemispheres — provides the other boundary-condition member of the curved-domain pair

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every integer side count $3\leq n\leq30$ in both determinant and natural-logarithm conventions for circumradius-one polygons; this contiguous prefix covers low and moderate polygonal symmetries, including nonconstructible polygons, before extending deeply toward the disk limit. The endpoint 30 is a noncanonical practical cutoff, not an application claim for each polygon. The angles and special-function arguments are determined by $n$)

How they were obtained:

The finite corner formulas of [1] and [2], combined with the Neumann/Dirichlet ratio in [3], are evaluated with exact rational coefficients and Arb ball arithmetic at 513 bits. The 56 values retain at least 499 relative accuracy bits. Every serialized 100-significant-digit interval contains a fresh 750-bit enclosure.

An independent computation uses the boundary-corner integral in [1], equations (18) and (20), at 150 and 200 decimal digits with 40 guard digits, analytic cancellation near zero and an explicit tail bound. This quadrature agreement corroborates the formulas; the finite Arb calculation supplies the certification. Separate square and triangle spectral products, with rigorous geometric tail bounds and the additional Neumann axis modes, check the normalization and zero-mode convention. Nonzero controls and deliberate perturbations test the comparisons.

The attached generator, checks, argument-selection note, derivation and numerical evidence describe the calculation. After readback, all 56 stored intervals pass fresh enclosure and independent integral comparisons; all determinant/logarithm pairs and all 28 ratios to the Dirichlet companion agree. The square and triangle spectral checks are repeated on their stored intervals.