Later it was noticed that this claim translates to a true statementabout the Riemann zeta function, with which Ramanujan was unfamiliar.)Ramanujan's innate ability for algebraic manipulations equaled or surpassedthat of Euler and Jacobi.
He introduced the Hilbert-Pólya Conjecture that theRiemann Hypothesis might be a consequence of spectral theory(in 2017 this Conjecture was partially proved by a team ofphysicists, and the Riemann Hypothesis be closeto solution!).
He was so prolific and original that some of his workwent unnoticed (for example, Weierstrass became famous for showinga nowhere-differentiable continuous function;later it was found that Riemann had casually mentioned onein a lecture years earlier).
Gauss selected "On the hypothesesthat Lie at the Foundations of Geometry" as Riemann's first lecture;with this famous lecture Riemann went far beyond Gauss' initial effortin differential geometry, extended it to multiple dimensions, andintroduced the new and important theory of differential manifolds.
Riemann's other masterpieces include tensor analysis,the theory of functions, and a key relationship betweensome differential equation solutions and hypergeometric series.
This work performs comparative study for three representative works of shadow detection methods each one selected from different category: the first one based on to derive a 1-d illumination invariant shadow-free image, the second one based on a hypothesis test to detect shadows from the images and then energy function concept is used to remove the shadow from the image.
Turing also worked in group theory, numerical analysis,and complex analysis; he developed animportant theorem about Riemann's zeta function;he had novel insights in quantum physics.
Weil proved a special case of the Riemann Hypothesis; he contributed,at least indirectly, to the recent proof of Fermat's Last Theorem;he also worked in group theory, general and algebraic topology,differential geometry, sheaf theory, representation theory, andtheta functions.
He also did seminal work with Riemann's zeta function,Dedekind's zeta functions,transcendental number theory, discontinuous groups,the 3-body problem in celestial mechanics,and symplectic geometry.
He worked with the Prime Number Theorem and Riemann's Hypothesis;and proved the unexpected fact that Chebyshev's bias, and, while true for most, and all butvery large, numbers, are violated infinitely often.
In addition to simpler proofs of existing theorems, new theorems by Landauinclude important facts about Riemann's Hypothesis;facts about Dirichlet series;key lemmas of analysis;a result in Waring's Problem;a generalization of the Little Picard Theorem;a partial proof of Gauss' conjecture about the density of classesof composite numbers;and key results in the theory of pecking orders, e.g.
Lie's work wasn't properly appreciated in his own lifetime, butone later commentator was "overwhelmed by the richness and beautyof the geometric ideas flowing from Lie's work."
Darboux did outstanding work in geometry,differential geometry, analysis, function theory,mathematical physics, and other fields,his ability "based on a rare combination ofgeometrical fancy and analytical power."He devised the Darboux integral, equivalent to Riemann's integralbut simpler;developed a novel mapping between (hyper-)sphereand (hyper-)plane; proved an important Envelope Theorem in thecalculus of variations; developed the fieldof infinitesimal geometry; and more.
(Asked what he would first do, if he were magically awakened aftercenturies, David Hilbert replied "I would ask whetheranyone had proved the Riemann Hypothesis.")ζ(.) was defined for convergent cases in Euler's mini-bio,which Riemann extended via analytic continuation for all cases.
He did very important work with prime numbers,proving that there is always a primebetween any and ,and working with the zeta function before Riemann did.