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{"Title": "Dirichlet spectral determinants of regular polygons", "Definition": "For the regular polygon $P_n$ of circumradius one, $D_n^D$ is the zeta-regularized determinant of the nonnegative scalar Dirichlet Laplacian $-\\partial_x^2-\\partial_y^2$ CITE{AS}. The positive determinant $D_n^D$ and its natural logarithm $L_n^D$ are one number in two invertible conventions.", "Parameters": {"n": {"type": "Z", "title": "number of sides", "display": "$n$", "constraints": "$n\\geq3$"}, "convention": {"type": "Symbolic", "title": "determinant or natural logarithm", "display": "convention", "values": {"det": "$D_n^D$", "logdet": "$L_n^D$"}}}, "Tags": ["analysis", "special values", "polygon", "functional analysis"], "Data properties": {"type": "R", "rigour": "proven", "complete": "no", "digits": "100", "complete-note": "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 $n$", "rigour details": "The finite formulas of CITE{AS} and CITE{Kalvin2020} are evaluated with exact rational coefficients and Arb ball arithmetic at 513 bits, retaining at least 498 relative accuracy bits over the selected entries. Every serialized 100-significant-digit interval contains a fresh 750-bit enclosure. The attached checks independently evaluate the boundary-corner integral in CITE{AS}, equations (18) and (20), at 150 and 200 decimal digits with 40 guard digits, using an analytically cancelled expansion near zero and an explicit tail bound. This quadrature agreement corroborates the formulas; it does not supply the certification, which comes from the finite Arb calculation. Independently separated square and triangle spectra give convergent sums with rigorous geometric tail bounds. Nonzero controls and deliberate perturbations test the comparisons. The generator, selection rationale, derivation and numerical evidence are attached. All 56 stored values also pass independent integral comparisons, enclosure containment and determinant/logarithm conversion checks after readback; the square and triangle spectral checks are repeated on their stored intervals."}, "Display properties": {"number-header": "$D_n^D\\text{ or }L_n^D$"}, "Numbers": {"3": {"det": "0.4562137291871239671588175396030622768725618053399871437714998800100700935964088552431970752939245803", "logdet": "-0.7847938749176002640207523311030596452895063902285088575311281767263908297931818251428533571059926259"}, "4": {"det": "0.4567895690780584107036212967006110448509819581967345167098482656329540046860045569816444916797900694", "logdet": "-0.7835324556688769647171162858468857319422187797040698228464202976835809031848864759319044585327361178"}, "5": {"det": "0.4614369140033657741157728037836941877733384603007759368539836022367244041603752101338281015196997735", "logdet": "-0.7734099321007320177679083218853627426312331300821242156996186333766464802047770058398083622325605871"}, "6": {"det": "0.4644606856681770336377122152001527119923333105040717674794516847261801660832318847795433931405878065", "logdet": "-0.7668783622689526630372991516100507663869043874745704817737119004325517534060850292082302382097164745"}, "7": {"det": "0.4663599784124541771121564018134633413492122699749782316149798879314802238909071996020019132569566840", "logdet": "-0.7627974572064930319746811139417180076577582317906010414273940714292314153493917854854731695296416373"}, "8": {"det": "0.4675737530567204652261584645123215380761627354450624017568120513751762753872305473824502242943926485", "logdet": "-0.7601981821471474619904735022400121829657179065249399206444113685074852409868956329906116676809316632"}, "9": {"det": "0.4683608974798595986270399080688793883783866649068029826456265703688610480270358303823061167593275780", "logdet": "-0.7585161317854552391320003530040511930097445164163537389626226962663964146348599804604812393125234669"}, "10": {"det": "0.4688733008228825927257932243843140127459275630393132362766928382191364221472099912003010941051480564", "logdet": "-0.7574226945314506726473784322168568293393283560097342047735417394727614873512034077444923976641623315"}, "11": {"det": "0.4692031658227387247540846800159814221032658712908325842398401994940811126252932631641286861263947071", "logdet": "-0.7567194149477901627332796303898537112534591745193381586074400731159499332160832283515982208431922795"}, "12": {"det": "0.4694085669944096952433761818397187281319205903366711538346634549911213405709402842924348716349517003", "logdet": "-0.7562817447849992410911236418897935477673150005506929609934086486282666937125871587633109170623609181"}, "13": {"det": "0.4695273186876276910812377520573737644749225278802436755581837950331888185849408450355225711049246475", "logdet": "-0.7560287952584936820157676573961217649114625177778212320768618247077467770732309236713969916087972404"}, "14": {"det": "0.4695847590296419363296795031569054402516319251857716896124126469678009586050746255744533123966693459", "logdet": "-0.7559064662129009613652865114201316236918177233127991381535853550782065800298209756000769830328952022"}, "15": {"det": "0.4695982863980014106048763706572887253029495340442043891373564717819558998458672509232068653408651213", "logdet": "-0.7558776595421965296651828405509063076204969446463288072641945244132852066137877707698063064449728157"}, "16": {"det": "0.4695800985115511740926204224109018392552828791815989809854265974589141377397354620986176712311549395", "logdet": "-0.7559163910265337927575556850819022834356421650742717796687459080815767368876616968313624089557296841"}, "17": {"det": "0.4695388997667674275614598311706307352886499040910306483608507164636140832633178869847657387581150950", "logdet": "-0.7560041301626511654884289914084913291494472696012311492790834940650106679751437295351299402707442449"}, "18": {"det": "0.4694809959456136097048113266164302573010475067232438780921278615987278442057535396518739920577664395", "logdet": "-0.7561274583721845361824328772647547718850612000033182360227846400121219244830285932755505664925060936"}, "19": {"det": "0.4694110138830500081941122493807926441240592883075548838866042991809348103781416637247492209895716097", "logdet": "-0.7562765320930397667738767620489729996957344639601847370627051662476409031059664461121155700786627659"}, "20": {"det": "0.4693323851770759634794980447650420057016315568197227463688565933467348309938313572372190895019968823", "logdet": "-0.7564440511535980524898944438877483865324137251078846992608214845894359533057966878132813787481174682"}, "21": {"det": "0.4692476777702919373503090167098964091610656409211574957219129714050456267126319687350261071939202804", "logdet": "-0.7566245523390760602457630228045815239990010355724069772283548942312519730440719340931502066986615093"}, "22": {"det": "0.4691588272771260915343086035362852575735753481971832248048509930674059511531882948952254110477810491", "logdet": "-0.7568139169544180294263552837958941054202148540952064844660958659756172457517327025109294183543388106"}, "23": {"det": "0.4690673009286294493216335574096871105755957522290934450454943074856301158562148135993105819231032258", "logdet": "-0.7570090220490970960738972208757047995883057915466449805486064344380160185781221531759887138624109710"}, "24": {"det": "0.4689742154173415053577471009256204837998839869179075660721099029894637787194670949984019765299633822", "logdet": "-0.7572074898353761011022554895954313863713876731272891352869687986502404971039160824859936855017646540"}, "25": {"det": "0.4688804226920052936798365426567445200747482459396981857400979531998776908754845396033112478974071806", "logdet": "-0.7574075053214390212000839995673155306858276263653281976562642037031282070833665258830933752194193397"}, "26": {"det": "0.4687865731426546226489801673397945598313660736165367204159311737194731883902409982975141774372294925", "logdet": "-0.7576076820367995020021144887555344683943039176569647550776881362839983010894716379207777844891960884"}, "27": {"det": "0.4686931626225911616852747882198654058149209802216433871019694754633463773485016843399117199171476681", "logdet": "-0.7578069621210700013229718435773278457380996658017589256853826352334520361566701620385129290572179749"}, "28": {"det": "0.4686005677746965695692397770108914902239607552802277774051896718343604300164292219849976456356317596", "logdet": "-0.7580045412687648589339332534119393675859843544604673602536741610427106171112922071699555008750534804"}, "29": {"det": "0.4685090728001945076745027100700339638859990521365446411940638752289875396717695574635078040078735866", "logdet": "-0.7581998118557511631364163549507388505345394453749749137233379215879110375074744840138211273629236181"}, "30": {"det": "0.4684188899018064466901867854296584268727919792315468754194104711528365657856776904517206541298774373", "logdet": "-0.7583923195025642475830680246041138521541969986498801150982841481319966074056472392230983883720888711"}}, "Comments": {"geometry": "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 HREF{Area_of_the_regular_n-gon}[the regular-polygon area table].", "spectrum": "Dirichlet conditions hold on every side, with the Friedrichs selfadjoint realization and eigenvalues counted with multiplicity. Write $Z_n^D(s)=\\sum_{k\\geq1}\\lambda_k^{-s}$ for $\\Re s>1$ and continue it analytically to zero. There is no zero eigenvalue, mass term or area division. All logarithms are natural and real. HREF{Dirichlet_eigenvalues_of_the_classical_planar_domains}[The planar-domain spectrum table] gives individual eigenvalues; its square and equilateral triangle have side length one, whereas its higher regular polygons have circumradius one.", "proof": "The polygon formula is Proposition 6 of CITE{AS}; CITE{Kalvin2026}, Section 7, proves the polygonal determinant formula for bounded simply connected Euclidean polygons. These polygons have flat smooth interiors, straight sides and positive angles summing to $(n-2)\\pi$, as required there.", "scale-notation": "The auxiliary scale $R_n$ in CITE{formula-scale-factor} is the reciprocal of HREF{Interior_conformal_radius_of_regular_Euclidean_and_ideal_hyperbolic_polygons}[the interior conformal radius] $r(P_n,0)$ in its Euclidean convention. It is not the physical circumradius, which is fixed at one."}, "Formulas": {"formula-determinant": "$L_n^D=-(Z_n^D)^{\\prime}(0),\\qquad D_n^D=\\exp(L_n^D),\\qquad L_n^D=\\log D_n^D$ CITE{AS}.", "formula-scale-factor": "$\\displaystyle c_n=Z_n^D(0)=\\frac{n-1}{6(n-2)},\\qquad R_n=\\frac{\\Gamma(1+1/n)\\Gamma(1-2/n)}{\\Gamma(1-1/n)}$, where $\\Gamma$ is the HREF{Values_of_the_Gamma_function_at_rational_numbers}[Gamma function] CITE{AS}.", "formula-polygon": "$\\displaystyle L_n^D=2c_n\\log R_n+\\frac{\\log n}{3(n-2)}-nF(\\alpha_n)$ CITE{AS}, equations (23) and (24).", "formula-corner": "$\\displaystyle F(a)=\\zeta_B^{\\prime}(0;a,1,1)-a\\zeta_R^{\\prime}(-1)+\\frac{a-a^{-1}}{12}\\log2-\\frac{a-1}{4}\\log(2\\pi)$ 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 HREF{Values_of_the_derivative_of_the_Riemann_zeta_function_at_rational_numbers#-1}[$\\zeta_R^{\\prime}(-1)$] is its derivative at $-1$ CITE{Kalvin2020}, equation (3.15).", "formula-rational-corner": "For coprime positive integers $p,q$, $\\displaystyle \\zeta_B^{\\prime}(0;p/q,1,1)=\\frac{\\zeta_R^{\\prime}(-1)}{pq}-\\frac{\\log q}{12pq}+\\left(\\frac14+S(q,p)\\right)\\log\\frac qp+\\sum_{k=1}^{p-1}\\left(\\frac12-\\frac kp\\right)\\log\\Gamma\\!\\left(\\left\\{\\frac{kq}{p}\\right\\}\\right)+\\sum_{j=1}^{q-1}\\left(\\frac12-\\frac jq\\right)\\log\\Gamma\\!\\left(\\left\\{\\frac{jp}{q}\\right\\}\\right)$, where $\\{x\\}=x-\\lfloor x\\rfloor$ and $S(q,p)=\\sum_{j=1}^{p-1}(j/p-1/2)(\\{jq/p\\}-1/2)$. Empty sums are zero CITE{Kalvin2020}, Lemma A.1.", "formula-dilation": "$\\displaystyle L^D(tP_n)=L_n^D-2c_n\\log t,\\qquad D^D(tP_n)=t^{-2c_n}D_n^D\\quad(t>0)$. Thus side length one uses $t=[2\\sin(\\pi/n)]^{-1}$, and area one uses $t=A_n^{-1/2}$ CITE{AS}.", "formula-controls": "$\\displaystyle D_3^D=\\frac{\\Gamma(1/3)}{\\pi^{2/3}3^{11/12}},\\qquad D_4^D=\\frac{\\eta(i)}{2^{3/4}}=\\frac{\\Gamma(1/4)}{2^{7/4}\\pi^{3/4}}$, where HREF{Dedekind_eta_values_at_rectangular_CM_points#1}[$\\eta(i)$] is the Dedekind eta value for the square lattice CITE{MR}, Corollaries 2.2 and 3.3."}, "References": {"AS": {"bib": "E. Aurell and P. Salomonson, On Functional Determinants of Laplacians in Polygons and Simplices, Communications in Mathematical Physics 165 (1994), 233\u2013260. Proposition 6 and equations (18), (20), (23)\u2013(25).", "arxiv": "hep-th/9304031", "doi": "10.1007/BF02099770"}, "Kalvin2020": {"bib": "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": "1910.00104", "doi": "10.1016/j.jfa.2020.108866"}, "Kalvin2026": {"bib": "V. Kalvin, Determinants of Dirichlet and Neumann Laplacians on Constant Curvature Surfaces with Geodesic Polygonal Boundary, preprint, version 1, 26 October 2026. Section 7.", "arxiv": "2610.35383"}, "MR": {"bib": "G. M\u00e5rdby and J. Rowlett, Spectral Invariants of Integrable Polygons, Journal of Fourier Analysis and Applications 31 (2025), article 69.", "doi": "10.1007/s00041-025-10202-6"}}, "Links": {}, "Keywords": ["regular polygon", "zeta-regularized determinant", "functional determinant", "spectral zeta", "log determinant", "Dirichlet Laplacian"], "Similar tables": [{"table": "HREF{Dirichlet_eigenvalues_of_the_classical_planar_domains}[Dirichlet eigenvalues of classical planar domains]", "relation": "holds individual eigenvalues, including circumradius-one regular polygons; the determinant regularizes the product of the entire positive spectrum"}, {"table": "HREF{Interior_conformal_radius_of_regular_Euclidean_and_ideal_hyperbolic_polygons}[Interior conformal radii of regular polygons]", "relation": "its Euclidean values are the reciprocals of the Schwarz\u2013Christoffel scale $R_n$ used in the determinant formula"}, {"table": "HREF{Logarithmic_capacity_of_regular_polygons}[Logarithmic capacity of regular polygons]", "relation": "holds the exterior conformal invariant of the same circumradius-one domains"}, {"table": "HREF{Dirichlet_spectral_determinants_of_Laplacians_on_hemispheres}[Dirichlet spectral determinants on hemispheres]", "relation": "uses the same zeta regularization and Dirichlet boundary condition on curved domains"}, {"table": "HREF{Neumann_spectral_determinants_of_Laplacians_on_hemispheres}[Neumann spectral determinants on hemispheres]", "relation": "uses the Neumann boundary condition on curved domains, with its constant zero mode omitted"}], "Programs": {"program-sage": {"language": "Sage", "code": "# Download the attached generate.py into the current directory.\nfrom generate import PolygonDirichletDeterminants\ng = PolygonDirichletDeterminants()\nfor params in g.enumerate():\n print(params, g.value(params, 100))\n"}}}