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{"Title": "Neumann spectral determinants of Laplacians on hemispheres", "Definition": "The zeta-regularized determinant of the unit round hemisphere's scalar Neumann Laplacian, omitting its constant zero mode CITE{Dowker}. The determinant $D^N_d=\\det{}^{\\prime}\\Delta_N$ and its natural logarithm $L^N_d=\\log D^N_d$ are one number in two invertible conventions.", "Parameters": {"d": {"type": "Z", "display": "$d$", "title": "dimension of the hemisphere", "constraints": "$d\\geq1$"}, "convention": {"type": "Symbolic", "display": "convention", "title": "determinant or natural logarithm", "values": {"det": "$D^N_d$", "logdet": "$L^N_d$"}}}, "Tags": ["special values", "analysis", "functional analysis"], "Data properties": {"type": "R", "digits": "100", "rigour": "proven", "complete": "no", "complete-note": "it holds every integer dimension $1\\leq d\\leq20$ in both determinant and natural-logarithm conventions for the radius-one scalar Neumann problem with its constant zero mode omitted; this noncanonical practical prefix covers the interval, low-dimensional hemispheres and ten dimensions of each parity while keeping an independent spectral continuation feasible, rather than extending only the high-dimensional range", "rigour details": "The finite sphere and Dirichlet formulas of CITE{CF} were recomputed with exact combinatorial coefficients and Arb ball arithmetic, and their log determinants were subtracted using the spectral decomposition of CITE{Dowker}. No rounded database values enter the computation. At 150 working decimal digits, all 40 values retain at least 504 bits of relative accuracy and overlap recomputations at 200 digits; 100 significant digits are stored. A separate direct continuation of the Neumann multiplicity-weighted spectral sum, with an exact geometric tail bound included in its Arb enclosure, agrees in every dimension and retains at least 560 bits of relative accuracy. Nonzero interval and low-dimensional controls, zeta functional-equation checks, changed tail split points, and deliberate perturbations test the comparisons. Every stored decimal interval is checked against the independent enclosure, and the determinant product and logarithm sum identities are checked using values read from the database. The direct continuation resolves the conflicting three-dimensional expression on printed page 7 of CITE{Dowker}; the calculation is preserved in the attached derivation. The generator, selection rationale and numerical evidence are attached."}, "Numbers": {"1": {"det": "6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136", "logdet": "1.837877066409345483560659472811235279722794947275566825634303080965531391854520795389486597271908395"}, "2": {"det": "4.480706884083051472769819152578970124572257213683489429404394891300582270334913869667895592492946197", "logdet": "1.499780820605574600208169056891178925389470234308186980150196153197485095260414743213793499300971709"}, "3": {"det": "2.513461667539913506354081963449459501905534440035607856308615700410508465949105985811228846214245343", "logdet": "0.9216609534035716323572688056613764911969342031577038786159657272057695915690797535261996568529187616"}, "4": {"det": "1.936573341145783262660051050535362949974768164666039340131159035991269490008368375161709944162295889", "logdet": "0.6609200922974785986905638705078894352275978350663596247999258078079614936260623979436308523594864327"}, "5": {"det": "1.565600756194695541901049196815040252916793657812783022952515764630561827499099578997567080540090677", "logdet": "0.4482696201059333821749616412913298104320565034779909504601175287564594626368031614697466576873696848"}, "6": {"det": "1.330585599632418142217462745277892489593359743389149427029014851224239581001313882198020594621850867", "logdet": "0.2856191458053538817496785928319235168639040955642695128591007120307714133391927932130975473929774609"}, "7": {"det": "1.168808070707179737570142395406018907637705093264002254054690663591866023170440463774063399187816869", "logdet": "0.1559844865530139122985450734944940493414223224126278158590265139770289780123182901451762919459115774"}, "8": {"det": "1.038024294879871959028800170997614302889714238624900816054807344587121334687212136140056764163175848", "logdet": "0.03731918994173141475049211856270437470001804521751160475198668653132792771818653070517258951139122351"}, "9": {"det": "0.9454861089894831965807802409303630273500511466066309828809818113942771483435387647989371718457792757", "logdet": "-0.05605608270018662508608190210713554626229121099945632606805837395811628191712029615694283841538016598"}, "10": {"det": "0.8610170998324581971022796804730478909676576341233225239510134207751792490078194782189619344440207524", "logdet": "-0.1496409143190341141302028915946166685129690119499237735793332267347396134092398683011239271893255390"}, "11": {"det": "0.8003201795094489516749103302491918121476379507845093733575540852478772163117582881448528787946443277", "logdet": "-0.2227434069958156223221248340691459374500018556549992061736162971772423612640886060657294100993623968"}, "12": {"det": "0.7407549442146872793120730197699054876088353580226391505095690588811998351201692542361047528279751298", "logdet": "-0.3000854179443422687317432558478682673565883851614361914138652359531147037075804230379346718016133379"}, "13": {"det": "0.6975096160418753705619610784020695647356931200803930657675742684534916910160794219487574246147156712", "logdet": "-0.3602389789416837591879247397296991100725096640509542625529846889553629645929461883366537797387335510"}, "14": {"det": "0.6529922205380977750102441357836469489866235969465928071826458514503158846755781549662547736185249798", "logdet": "-0.4261900631945920296270743277351528888516987924133600226009922456417969806841883559188572592423957820"}, "15": {"det": "0.6204243818392869127662003210938526664035160355591595895198896164176731142777631937243225804003461672", "logdet": "-0.4773515482597114203071572994730491847962139694487802065615972080725000632213785786974584053368317033"}, "16": {"det": "0.5857482246793301612889676154943906707481437227313959813006492091570492353423971297669994628464259394", "logdet": "-0.5348652324493070630152609146147400540200132446269819122195748870237953935370856526397759987654670109"}, "17": {"det": "0.5602268136732349452395543206694874316793533357544540410297049191178047649443061020033831580587277240", "logdet": "-0.5794135528366994833004613915803563663928877687283088299200141619627812735038005603384556529747608066"}, "18": {"det": "0.5323667086592463292636083421249024256990431851210448543506219870189480461723981064896204559255721300", "logdet": "-0.6304227251089309759072361558760725401161690735993201278224180871902525929267672237353957468077906447"}, "19": {"det": "0.5117599058806279676302703349334373268367118096261278121524547630838762985804671239096910499505707056", "logdet": "-0.6698996977533795491650628790638888964952669353297770662176045739126018837357913101925349400115140435"}, "20": {"det": "0.4888306854461313875417235962198078067754710574652490360002434455904508269848541084366269404556685944", "logdet": "-0.7157390959958643194374764702713373131249992068349076990056938372905055794075698278136553512163397101"}}, "Keywords": ["hemisphere", "half sphere", "Neumann determinant", "zeta-regularized determinant", "functional determinant", "scalar Laplacian", "Laplace-Beltrami", "spectral zeta", "log determinant"], "Comments": {"geometry": "The hemisphere is $H^d=\\{x\\in\\mathbb{R}^{d+1}:|x|=1,\\ x_{d+1}\\geq0\\}$ with its induced round metric. The nonnegative operator is $\\Delta_N=-\\operatorname{div}\\operatorname{grad}$ on scalar functions, and the boundary condition is $\\partial_\\nu u=0$ on the equator, where $\\nu$ is the outward unit normal. For $d=1$, $H^1$ is an interval of length $\\pi$ with the Neumann condition at both endpoints. Radius is fixed to one, with no volume normalization or additive shift.", "regularization": "The spectral zeta function $Z^N_d(s)$ is continued meromorphically from $\\Re s>d/2$ and is regular at $s=0$. Powers of positive eigenvalues use real logarithms. The prime on $\\det{}^{\\prime}$ omits the single constant zero mode; the prime on $Z^N_d$ differentiates with respect to $s$. CITE{Dowker}, Section 2.", "conventions": "The positive determinant $D^N_d$ and its natural logarithm $L^N_d$ are one number in two invertible conventions, stored for the same dimensions.", "evaluation": "Equatorial reflection separates the spherical harmonics into the Neumann and HREF{Dirichlet_spectral_determinants_of_Laplacians_on_hemispheres}[Dirichlet] hemisphere sectors. Subtracting the Dirichlet log determinant from the HREF{Spectral_determinants_of_Laplacians_on_spheres}[sphere log determinant] gives $L^N_d$. The finite expressions in CITE{CF}, Theorems 2.7, 2.8 and 3.2, use HREF{Values_of_the_derivative_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta derivatives] and elementary terms. CITE{CFHTML}."}, "Formulas": {"spectrum": "$\\displaystyle Z^N_d(s)=\\sum_{k=1}^{\\infty}\\binom{d+k-1}{d-1}[k(k+d-1)]^{-s}$ for $\\Re s>d/2$, where $k$ is the spherical-harmonic degree. CITE{Dowker}, equations (1) and (2).", "conversion": "$D^N_d=e^{L^N_d}$ and $L^N_d=\\log D^N_d=-(Z^N_d)^{\\prime}(0)$.", "interval": "$Z^N_1(s)=\\zeta(2s)$, $D^N_1=2\\pi$ and $L^N_1=\\log(2\\pi)$, where $\\zeta$ is the HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta function].", "low-dimensions": "$L^N_2=-\\zeta^{\\prime}(0)-2\\zeta^{\\prime}(-1)+\\tfrac14$ and $L^N_3=\\tfrac12\\log(\\pi/2)-\\zeta^{\\prime}(-1)-\\zeta^{\\prime}(-2)+\\tfrac12$. These follow from the sphere and Dirichlet formulas of CITE{CF}, Theorems 2.7, 2.8 and 3.2.", "sphere-splitting": "$Z^{S^d}(s)=Z^D_d(s)+Z^N_d(s)$, $D^{S^d}=D^D_dD^N_d$ and $L^{S^d}=L^D_d+L^N_d$. Here the superscript $S^d$ denotes the HREF{Spectral_determinants_of_Laplacians_on_spheres}[unit round sphere with its zero mode omitted], and $D$ denotes the HREF{Dirichlet_spectral_determinants_of_Laplacians_on_hemispheres}[Dirichlet hemisphere] with the same radius and scalar operator. The quantities $Z$, $D$ and $L$ denote the spectral zeta function, determinant and natural log determinant, respectively. CITE{Dowker}, equation (3)."}, "Similar tables": [{"table": "HREF{Dirichlet_spectral_determinants_of_Laplacians_on_hemispheres}[Dirichlet spectral determinants of Laplacians on hemispheres]", "relation": "uses the same hemispheres and scalar Laplacians with the Dirichlet boundary condition"}, {"table": "HREF{Spectral_determinants_of_Laplacians_on_spheres}[Spectral determinants of Laplacians on spheres]", "relation": "holds the full-sphere determinants, each equal to the product of the Dirichlet and Neumann hemisphere determinants"}, {"table": "HREF{Neumann_spectral_determinants_of_homogeneous_oscillators}[Neumann spectral determinants of homogeneous oscillators]", "relation": "uses spectral zeta regularization for half-line Schr\u00f6dinger operators with the Neumann boundary condition"}], "References": {"Dowker": {"bib": "J. S. Dowker, Functional determinants on spheres and sectors, 1993, Section 2.", "arxiv": "hep-th/9312080"}, "CF": {"bib": "Jos\u00e9 Cunha and Pedro Freitas, Recurrence formulae for spectral determinants, 2024, Theorems 2.7, 2.8 and 3.2, and Appendix A.", "arxiv": "2404.12114v1"}}, "Links": {"CFHTML": {"title": "Cunha and Freitas: full text and finite determinant formulas", "url": "https://arxiv.org/html/2404.12114v1"}}, "Display properties": {"number-header": "$D^N_d\\text{ or }L^N_d$"}, "Programs": {"program-sage": {"language": "Sage", "code": "# Download generate.py and finite_formulas.py into the current directory.\nfrom generate import logdet\nfor d in range(1, 21):\n L = logdet(d, working=150)\n print(d, L.exp(), L)\n"}}}