i have only heard of selberg's trace formula, but he is famous, much more as a mathematician than erdos, who is known primarily for being eccentric, and for writing a huge number of papers.
in fact there is an invariant known as the "erdos number" which is one less than the smallest number of people say A,B,C,D...,X, such that A = you, X = erdos, and A has written a paper with B, and B has written a paper with C and so on.
so it is the smallest link between you and erdos in terms of writing papers. I.e. if you have written a paper with erdos, then your erdos number is 1. If you have not, but you have written a paper with someone who has written a paper with erdos, then your erdos number is 2, and so on. Erdos wrote so many papers that most people have a fairly small erdos number. It may be conjectured that everyone in the world who has written a paper has a finite erdos number.
I for example who have very little interest in or admiration for erdos, and work in a different area, nonetheless have an erdos number of about 3.
erdos has perhaps the record for the largest number of papers, about 1,000 or so.
but to me this is almost a joke, as no one can write that many good papers. or it would be except that i know from personal experience that erdos has had a very good influence on young people in interesting them in mathematics with his many problems. so he is doing this (or was) for a good unselfish reason. i.e. he was not just publishing papers to make money or become famous. still the quality of the output is highly questionable in many cases.
for example riemann wrote far, far, fewer papers, his collected works list 31, of which i am personnaly familiar only with 5.
These 5, the only ones by riemann that are famous (to me) are his thesis on riemann surfaces where he introduces concepts of topology to study plane curves and complex functions, his followup paper on abelian functions including his analysis of analytic functions on a riemann surface and the first part of the famous riemann - roch theorem, his beautiful paper on the vanishing of theta functions containing his (partial) proof of the famous riemann singularities theorem, his great paper (habilitationschrift) on differential geometry in which he defined n dimensional manifolds and curvature (translated in spivak's book on diff geom, volume 2) , and his famous paper on prime numbers.
yet each of these 5 is absolutely Earth shattering, and his prime number conjecture, the so called "riemann hypothesis", is perhaps the most famous unsolved problem in mathematics today.
ironically, although one can readily buy a translation of almost any piece of trivial #%**&* written in any language, I do not know of a translation into english of any other of the great papers of riemann except that in spivak's book. (nor of galois' famous letter. perhaps that is why so few people know it anticipated some of riemann's works, and in particular that it contains more than the theory of groups.)
the only mathematics i know of that erdos is famous for is his "elementary" proof of the prime number theorem. "elementary" means that someone else proved it first (hadamard?) using more sophisticated mathematics and erdos proved it afterward using fewer tools. this does not in itself impress me, as often the real insight into a theorem comes from using more sophisticated and more natural tools suited to the problem. It is often easy to analyze someone else's proof and then remove the sophisticated techniques in an unnatural way, so that they become disguised, and claim to have an elementary proof, but a proof no one would ever have thought of without the original proof to guide them.
a friend of mine once explained to me a small modification of a problem of erdos on the other hand which I solved in 5 minutes, and everyone else i have told about it solved it in a few seconds. so i know some of his problems are rather easy.
to be fair, erdos also posed some extremely non trivial problems that lasted for years, and some of my most respected colleagues are proud to have solved them, but i am still not moved too much, perhaps because of that first experience.
even his hard problems do not seem super interesting to me, because the ones i have seen are somewhat narrow in scope and application.
i will admit however that i know some fine mathematicians who have the highest possible regard for erdos, but not huge numbers of them. almost everyone on the other hand reveres the names of riemann, gauss, euler, hilbert, and archimedes. i see on looking back that my list overlaps considerably with the first post on this thread, especially since i omit physicists, so einstein is not eligible for my list. i would also admit Newton but do not want to give up any of mine, so i have 6.
in the 20th century i still like grothendieck, serre, deligne, weil, weyl, poincare, hilbert, chern, kodaira, milnor, mumford, faltings, witten (a physicist who definitely impacts mathematics) and others.
I do not know if these are comparable to the ancients, with less than 100 years of perspective. If so, then the 20th century may be the richest source of great math scientists, and it probably is. my personal favorite century for rich math though is the 19th, led by riemann and gauss.
the early 21st century may lag somewhat behind the 2oth, since today's NY Times records that congress decided to cut the budget of the national science foundation, partly in order to have money to fund the rock and roll hall of fame in cleveland and the country music hall of fame in nashville. Is it any wonder why the US lags the world in math/science, but not in football or rock and roll?
