Bernhard Riemann

Record year1868

M rationale6.7 is above the general disciplinary-foundation centre because multidimensional metric, local curvature, and empirical boundaries jointly reconfigured the problem space of geometry; a complete modern axiomatic and operational system had not yet formed, and later transfer into physics is not included, so the score does not enter 8.

attribution rationaleIn the 1854 lecture, Riemann introduced multidimensional metric, local curvature, and empirical boundaries of physical measurement, taking the main share of mathematical content; posthumous editorial publication receives separate credit.

Record year1859

M rationaleAnalytic continuation, the functional equation, the xi function, and explicit inversion form a systematic connection across generations and thus belong to the disciplinary-foundation tier; the original text's incomplete proofs, the qualification of summing zeros, and the RH, prime number theorem, and successor L-function results are all excluded.

attribution rationaleRiemann connected analytic continuation, the functional equation, the xi function, zeros, and prime-counting inversion into a method, taking the principal synthesis and derivation share; unfinished conjectures and successor theorems are not included.

Record year1851

M rationaleThe complete 1851 framework changed the objects that could be handled and their methodological relations, reaching the global disciplinary-foundation tier; unresolved general proofs and later modern systems are not claimed. This item is assessed at M6.3.

attribution rationaleRiemann connected complex functions, branched surfaces, connectivity cuts, and conditional conformal constructions into a framework; the principal credit is for this synthesis and construction, while the specific predecessor methods used have separate shares.

Record year1854; published posthumously

summaryIt organized the definition of the definite integral through partitions, sampling, and a fixed limit, and, together with integrability, conditions for trigonometric-series representation, and counterexamples, constituted a foundational achievement in analysis.

M rationale6.1 is above an ordinary field change because the integral definition and trigonometric-series counterexamples jointly shaped rigorous analytical capability; subsequent integration theories and more complete criteria are excluded, so it does not extend into the upper part of band 7.

attribution rationaleRiemann completed interconnected research on the integral definition, integrability, and trigonometric series, credited as one whole; Dirichlet's and Fourier's problem and methodological inputs, and Dedekind's posthumous publication, are accounted for separately.

overlap boundaryThe research on the integral definition and trigonometric series is counted once as a combined item; subsequent integration theories and general criteria are excluded.

Record year1857

M rationaleAfter stripping away the shared 1851 apparatus, the 1857 inversion and theta increment remains an internationally important methodological system; the available facts are insufficient to treat this increment alone as equivalent to the foundation of an entire discipline. 5.8 is adopted.

attribution rationaleRiemann extended inversion from special cases such as elliptic ones and constructed the 1857 Abelian and theta methods, bearing the principal share for the generalization and methodological construction; Abel supplied the starting point of elliptic inversion, Jacobi supplied materials on elliptic inversion and theta functions and the technical comparison in the original work, Dirichlet supplied the proof route via the variational principle explicitly used in the original work, and Gauss corresponds only to input on local complex representation. The residual account retains only inputs or responsibility gaps not yet resolved through item-by-item verification and full-text review; prior-work special cases or proof routes are not equated with Riemann's delivered general inversion.

Record year1859–1860

M rationaleCharacteristic invariants, hodograph linearization, and a judgment of compressive steepening changed the field's capacity to handle finite plane gas waves, exceeding a local model; the scope remains limited to smooth, nondissipative plane waves, so it is only slightly above the center of band 5.

attribution rationaleRiemann gave the characteristic invariants, linearization, and judgment of compressive steepening for smooth plane gas waves; the Euler equations and other adopted gas relations retain input shares.

Record year1857

M rationaleThe characterization of singularities, series, transformations, and conditional integral representations formed a unified interface for the second-order hypergeometric equation, hence slightly above the center of internationally important methods; general continuation and surface apparatus have already been claimed by the preceding item, and subsequent general existence theory is also excluded.

attribution rationaleRiemann organized the P-function, second-order equations, series, and integral representations through singularities, branches, and exponents; the principal credit is for this unified construction, while the prior results used are accounted for separately.

Record yearManuscript; arranged and published in 1932

summaryA numerical method for computing the zeta function on the critical line, consisting of a saddle point, a finite sum of length approximately sqrt(t/2π), and an asymptotic remainder; limited to the original method and the identifiable 1932 delivery.

M rationaleThe finite sum and asymptotic remainder form a practically computable method for the zeta function; the whole, as completed in the 1932 arrangement, adopts M5.2. Later computational improvements and zero verification are excluded.

attribution rationaleRiemann's manuscript left the methodological core of the saddle point, finite sum, remainder asymptotics, and added terms, receiving a 55% individual share; Siegel's deciphering, completion, and 1932 public write-up are credited separately.

overlap boundaryOnly the numerical computational method is counted; general analytic continuation, the functional equation, prime inversion, and later zero verification are not counted again.

Record year1857; published in 1876

summaryUnder conditions of finite singularities, replacement by constant monodromy, distinct local multipliers, and finite-order growth, a determinant argument yields linear relations with polynomial coefficients and an nth-order ODE.

M rationale4.8 approaches band 5 because the determinant mechanism and two corollaries form a reusable construction of general order; it does not solve arbitrary-monodromy existence, resonance, or completeness at infinity, and therefore does not enter band 6.

attribution rationaleRiemann's manuscript derived linear relations and two corollaries from finite singularities, constant single-valued transformations, finite-order growth, and a determinant argument, bearing the principal constructive share; incomplete general existence theory is not counted.

overlap boundaryOnly the general-order construction under finite conditions is counted; explicit second-order hypergeometric solutions belong to existing results, and subsequent general existence theory is excluded.

Record year1866

summaryA later increment that treated the identically-zero exception in existing theta inversion theory, giving vanishing conditions and related corollaries on derivative conditions.

M rationale4.3 reflects the substantive repair that vanishing and derivative conditions made to the existing framework; it is not the whole of modern singularity theory and does not duplicate the main value of 1857, so it is below the middle of band 5.

attribution rationaleRiemann treated the identically-zero exception in the 1857 framework and gave corollaries on vanishing and derivative conditions, directly bearing the principal incremental share; the earlier framework is not credited again.

overlap boundaryOnly the new research on the identically-zero exception and vanishing conditions is counted; the main 1857 inversion achievement is not duplicated.

Record year1860—1861

summaryUnder the assumption of linear motion, a systematic analysis of the axes of rotation, relative internal motion, constant shape, and restricted variational stability of a homogeneous self-gravitating fluid ellipsoid.

M rationale4.4 is above a general result in a specialist subfield because motion, shape, and stability are linked into a systematic method; unconditional nonlinear stability, exact complete integrals, and subsequent results are excluded, keeping it below the middle of band 5.

attribution rationaleUnder the prior assumption of linear motion, Riemann completed the analysis of principal axes, relative motion, constant shape, and restricted stability, bearing the principal share; predecessor assumptions and known special cases are counted separately.

overlap boundaryOnly ellipsoid analysis under the condition of linear motion is counted; it is not extended to arbitrary fluids or unconditional nonlinear stability.

Record yearManuscript; arranged and published in 1932

summaryFrom the Φ identity, an exact double oblique-contour integral representation of the ζ-function is obtained, which on the critical line becomes the real part of a single integral with a Gamma factor.

M rationaleThe exact double-contour representation and its usable form constitute an independent computational tool; M4.6 is adopted. Its scope is narrower than the whole theory of the ζ-function and does not duplicate analytic continuation or the functional equation.

attribution rationaleThe mathematical core of the exact integral representation comes from Riemann's manuscript and receives a 55% individual share; Siegel's substantial rewriting for application and written presentation retains independent credit.

overlap boundaryThe exact integral representation and finite-sum asymptotic computation are scored as separate increments; the shared identity is not made into another item.

Record year1862 course; published in 1876

summaryOn a nonsingular nonhyperelliptic genus-three plane quartic, it constructs a finite configuration of 28 bitangents using odd theta characteristics, linear forms, paired zeros, and a grouping into six pairs.

M rationale4.3 is above an ordinary special case because the 28 objects, grouping of characteristics, and algebraic relations form a complete finite configuration; it is a specialized application of a general framework, so it does not enter the upper part of band 5.

attribution rationaleIn a course, Riemann completed the construction of theta characteristics, bitangent pairings, and groupings for genus-three quartics; Röhr's lecture notes and subsequent editorial publication retain their respective credit.

overlap boundaryOnly the specific construction of quartic bitangents is counted; the general theta framework, teaching service, and modern extensions are not duplicated.

Record year1876

M rationale4.1 reflects a field-level increment formed jointly by a new concept and restricted relations; defective notation, editorial corrections, and incomplete general theory keep it below the middle of band 5.

attribution rationaleRiemann's manuscript proposed dimension-distinguished closed cycles that are not boundaries, connectivity, and restricted relations of variation, bearing the principal share for the concepts and relations; editorial corrections and publication are counted separately.

Record year1847; published in 1876

summaryA fractional-order integral kernel with a finite starting point, a restricted construction extended from ordinary differentiation, and computations for ordinary power functions.

M rationaleThe finite-starting-point kernel and operations on order form a reusable family of methods, deeper than an isolated formula; unproven broad adoption cannot be used to advance international-platform value. The center is 4.0 for a specialist method.

attribution rationaleRiemann's manuscript gives the finite-starting-point integral kernel, restricted differential extension, and power-function computations, and thus bears the principal constructive share; prior ideas and posthumous editorial publication retain credit separately.

overlap boundaryRelated fractional-order formulas are combined into one family of methods; it is not credited as inventing all fractional calculus or modern applications.

Record yearAround 1860; published posthumously

summaryA conditional construction for minimal surfaces with circular sections in parallel planes, including the catenoid and a new one-parameter family, arranged and made public by 1868.

M rationale4.0 is at the threshold of a field-important result because the delivery is not merely a single example but a condition-controlled new one-parameter family; coverage is limited to parallel circular sections, so it does not exceed 5.

attribution rationaleThe core construction of minimal surfaces with circular sections comes from Riemann's original manuscript; posthumous arrangement and completion are credited to the actual participants, and later classification results are not read back into the original manuscript.

overlap boundaryIt is distinguished from the three-line boundary construction; existing complex-function tools, extensions to nonparallel circular sections, and later classifications are not duplicated.

Record year1861

M rationale3.4 reflects that six conditions and their relations of independence form a reusable criterion, with independent confirmation of historical effect; the scope does not extend to general dimensions or all heat problems, keeping it far from band 5.

attribution rationaleRiemann completed sufficient conditions and a concrete criterion for constant-coefficient transformations of the three-dimensional heat equation, bearing the principal share; the early geometrical framework is not duplicated.

Record yearPublished in 1876

summaryIt reduces the potential or steady-temperature problem of parallel cylinders to a planar boundary problem, and constructs conditional circular-boundary maps using mirror doubling, circle reflection, Möbius transformations, and ratios of solutions of second-order equations.

M rationaleConditional reduction, circle reflection, and construction by solution ratios form a complete specialized method; M3.5 is adopted. Determination of global constants and arbitrary circular-boundary configurations remain unclosed, limiting the scope.

attribution rationaleRiemann left the mathematical core of the formulas, conditional reduction, and construction by solution ratios; Weber made these materials into an intelligible public text, and his substantial editorial work is credited separately.

overlap boundaryOnly the specific construction for circular-boundary potential problems is counted; general complex functions, doubling, and differential-equation tools are not duplicated.

Record year1852; published in 1876

summarySpecific limiting relations obtained by integrating boundary series and approximation when an elliptic modular function approaches a rational real boundary radially in the period ratio.

M rationaleThe rational-boundary relation and radial approximation form a reusable finite method; M3.4 is adopted. Modern rigorous supplementation and broader modular-function theory are excluded.

attribution rationaleIn a 1852 fragment, Riemann completed specific operations of boundary-series integration and radial approximation. Credit includes the original-manuscript method, without assigning later rigorous supplementation to him; deciphering, correction, and publication are allocated separately.

overlap boundaryOnly the specific method at rational boundaries is counted; general theta inversion, vanishing conditions, and later rigorous supplementation are not duplicated.

Record year1859; published in 1870

summaryFor nondegenerate multivariable periodic functions, an upper bound of 2n period generators and a criterion that variables degenerate when the bound is exceeded.

M rationale3.2 is slightly above the center of band 3 because the result gives both a generation bound and a consequence of degeneration, forming a complete diagnostic tool; it does not incorporate 1857 inversion, all periodic theory, or later applications, so it does not approach band 5.

attribution rationaleIn an 1859 letter, Riemann completed the proof of the upper bound on period generation and the degeneration criterion, bearing the principal share; his own early framework is an existing tool and does not create a second result.

overlap boundaryOnly the 1859 increment on the period upper bound is counted; 1857 inversion and general existence theory are not duplicated.

Record yearBy 1876

M rationaleCoordinates, equations, and boundary selection form a continuously reusable method and are deeper than a single worked example; but it reuses potential-theory and hypergeometric tools, treats only toroidal geometry, and has neither experimental, industrial, nor general-potential-theory coverage, so it lies near band 3.

attribution rationaleRiemann completed the coordinates, equation transformation, separation, and selection of boundary solutions for toroidal geometry, bearing the principal share for the specific method; general potential-theory and hypergeometric inputs retain credit separately.

Record year1863

M rationale3.0 falls at the start of specialist-subfield-important results because the matching principle and application of hypergeometric quotients form a reusable method; the original manuscript is incomplete and convergence theory was completed by successors, limiting further elevation.

attribution rationaleRiemann's manuscript gives a matching method for finite asymptotic continued fractions and an application to hypergeometric quotients, bearing the principal share; subsequent convergence theory is not assigned to the manuscript.

Record year1858

M rationale2.9 approaches 3 because the preserved equation and fundamental potential are themselves complete and usable; the connected physical derivation contains major errors and lacks empirical closure, limiting the depth and coverage of the complete result.

attribution rationaleRiemann proposed the inhomogeneous wave equation and a retarded fundamental potential; the principal credit is limited to the valid mathematical structure, and the connected but erroneous physical derivation is not credited as a complete achievement.

Record yearPublished in 1876

summaryA specific construction of a minimal surface with a three-straight-line boundary, having two boundary vertices and an infinite sector.

M rationale2.6 lies between a result of enduring use to a small specialist community and a specialist-subfield-important result: the construction is complete and has a methodological increment, but its coverage is narrow and it reuses existing general representations and complex-function tools.

attribution rationaleRiemann completed a short construction linking two vertices, an infinite sector, and a spherical triangle, bearing the principal share; reused general tools are not counted as a new result here.

overlap boundaryOnly the specific construction with three straight-line boundaries is counted; the parallel-circle-section family and examples completed separately by editors are excluded.

Record year1855

M rationaleIt links deposition observations, a current boundary problem, and asymptotic corrections into a method that can help a finite specialist group; its scope is limited to a specific thin-layer axisymmetric apparatus, and complete experimental agreement or industrial coverage is unproven.

attribution rationaleRiemann completed the boundary-value model, finite-boundary corrections, and asymptotic solution for color rings; deposition laws, current relations, and original observations are explicit prior inputs and are not jointly credited to him.

overlap boundaryLimited to thin layers, finite boundaries, and asymptotic solutions; Faraday is only an explicit input for the law of proportional deposition.

Record year1864

summaryFor the gravity problem of a finite homogeneous elliptic cylinder or half-cylinder, a specific solution method that converts force components to an elliptic potential and selects an integral branch to satisfy boundary conditions.

M rationale2.1 is slightly above the strong-result band for one person or a small group because this is a publicly reusable analytical construction; specialized geometry and lack of facts about broader use keep it below the center of band 3.

attribution rationaleRiemann converted the specific elliptic-cylinder gravity problem into a potential function and an integral with a selected branch, bearing the principal constructive share; his existing complex-analysis tools are not made into another result.

overlap boundaryThe static gravitational solution for elliptic cylinders and the internal motion and stability of fluid ellipsoids are different achievements; general potential theory is not duplicated.

Record year1854

M rationale1.6 reflects that equations and comparison with phenomena form a reusable model; it makes only qualitative comparisons and lacks apparatus parameters, keeping the center below the middle of band 2.

attribution rationaleRiemann organized equations for charge relaxation and made qualitative comparisons, constituting the principal delivery; adopted electrical relations, observations, and corrections retain shares separately.

Record year1857—1862

summarySeveral identifiable batches of advanced-mathematics courses from 1857 to 1862, and the availability of lithographed lecture notes compiled and distributed from notes by an actual attendee.

M rationaleContinuous multiyear courses, several types of courses, and distributable notes form an actual teaching service; M1.4 is adopted. Coverage of student rosters, note contents, and interinstitutional adoption remains limited, so it is not valued as the training achievement of a whole generation of mathematicians.

attribution rationaleRiemann prepared and taught several advanced-mathematics courses, bearing the principal teaching share; the actual compilers of attendance notes and distributors of lecture notes receive credit separately.

overlap boundaryIt is distinguished from early courses by period; teaching service does not receive again the scores of the mathematics taught in the courses, and particular lecture-note compilation is not duplicated.

Record year1854—1856

summaryEarly courses actually taught from 1854 to 1856 on PDEs and physical applications and on Abelian functions, providing instructional help to about eight and a small number of named attendees.

M rationale0.7 is above one finite instance of help because the courses continued and covered two advanced topics; it does not anticipate students' subsequent achievements or duplicate paper knowledge, keeping the center below 1.

attribution rationaleRiemann undertook actual course preparation and teaching and therefore receives the principal share; explicitly adopted teaching models are counted separately, and students' later achievements are not added to the teacher.

overlap boundaryThe early courses are combined into one teaching service; later courses, papers, reviews, and students' later achievements are not duplicated.

Record year1865

summaryIn exchanges over several weeks in 1865, Riemann explained theta formulas to a researcher, proposed specific proof advice, and gave feedback on that person's proof approach.

M rationale0.38 is above a single short opinion because the exchange lasted several weeks and included explanation of formulas, proof advice, and feedback; it is also below the vicinity of the 0.5 anchor for an important multiyear beneficiary, avoiding anticipation of the beneficiary's later career.

attribution rationaleIn exchanges over several weeks, Riemann explained formulas, suggested proof methods, and gave feedback on Prym's proof approach, bearing the principal guidance share; this does not amount to independent verification that the proof was correct, nor does it claim Prym's subsequent achievements.

overlap boundaryResearch consultation over several weeks is distinguished from formal courses; theta theory itself, the recipient's papers, and subsequent dissemination are excluded.

Record year1866

M rationaleThe completed methodological section contains an executable sequence of explanation, alternative hypotheses, and empirical correction, exceeding generic advice and a single minor suggestion; but it does not close physiological mechanisms, experimental effects, or stable group adoption, and can only be judged substantive local help, not band 3 for a specialist subfield. The completed section has no demonstrated continuing use by a professional group; it recognizes actual knowledge service without treating abstract completeness as broad impact, with a local-help center of 0.45.

attribution rationaleRiemann's specific organization of methods for research on sensory organs is assigned 0.67; identifiable inputs from Herbart, Henle, and Schering are retained, and 0.05 remains unallocated. Newton's analytic–synthetic method proves only general prior influence and does not point to an independent new action in this item that has not been consumed by the original function.

overlap boundaryDelete Newton's generalized-prior share of 0.10; count only Riemann's completed methodological section, not an unfinished ear mechanism or modern biomedical benefits.

Evidence gapsExisting sources and a limited set of principal families are retained for the assessment; evidence of continued use, independent alternative paths, and predecessor contributions is not exhaustive. This revision did not reread the full texts of all original works.

Record year1861

summaryA substantive recorded paper review adopted by another committee member, together with historical explanation connected to the review of the same paper.

M rationaleThe review had specific technical content and was adopted by another committee member, exceeding procedural opinion; the known effect is limited to one review, with no demonstrated enduring or broad benefit, so M0.32 is adopted.

attribution rationaleRiemann submitted a specific technical review and historical explanation of Euler, and therefore receives more than half the share; Weber's adoption and committee disposition, and Hankel's response and use, receive separate credit, and Hankel's paper discovery is not counted.

overlap boundaryThe review and historical explanation of the same paper are counted once; Hankel's paper discovery, existing mathematical theorems, and the whole period of teaching are excluded.