I think it useful if one gives ones reasons for ranking someone highly, so here is my review of Riemann's collected works, in translation:
Review of Bernhard Riemann, Collected Papers, translated by Roger Baker, Charles Christenson, and Henry Orde,
published by Kendrick Press, copyright 2004, 555 pages.
My father's childhood copy of Count of Monte Cristo is inscribed: “this the best book I ever read,” exactly my opinion of this translation of Riemann's works. After the shock of how good and extensive these works are, by a man who died at 39, one is overwhelmed by his succinct, deep insights. It is amazing no English version of these works has appeared before, and this event should be celebrated by all mathematicians and students who read primarily English.
This translation contains all but one of the papers I-XXXI from the 1892 edition of Riemann’s works, but not the “Nachtrage”. The translation seems faithful, misprints are few, it reads smoothly, and the translators do not edit or revise Riemann's words, in contrast to the selections in "A source book in classical analysis", Harvard University Press.
I feared Riemann was obscure, and inconsistent with modern terminology, but once one starts reading, the beauty of his ideas begins to flow immediately. There is no wasted motion, computational results are written down with no visible calculation, and their significant consequences simply announced. This is a real treat. Mysterious statements become a pleasant challenge to interpret, in light of what they must mean. Even outmoded language is clear in context.
This is a concise and understandable source for subjects that paradoxically are harder to learn from books which expend more effort explaining them. That Riemann omits details, and knows just what to emphasize, make it a wonderful introduction to many topics. Even those I thought I understood, are stripped of superfluous facts and shine forth as simple principles.
Some highlights for me: "Riemann's theorem" and the "Brill - Noether" number, are both derived on page 99. If L(D) = {meromorphic functions f with div(f)+D ≥ 0}, on a curve of genus g, then dimL(D) - 1 = dim ker[S(D)], where S(D) is a (2g) by (g+deg(D)) “period matrix”. Hence (Riemann's theorem) deg(D)-g ≤ dimL(D) -1 ≤ deg(D), and C(r,d) = {divisors D with deg(D) = d and dimL(D) > r} has a determinantal description = {D: rank(S(D)) ≤ (d-r+g)}.
Hence a generic curve should have a non constant meromorphic function with ≤ d poles only if d ≥ (g/2) + 1, by the intersection inequality (d-1) ≥ (g+1-d) (= codimension of the rank (d-1+g) locus, in (2g) by (g+d) matrices). The similar estimate (d-r) ≥ r(g+r-d) gives the “Brill - Noether” criterion for C(r,d) to be non empty for all curves of genus g, 16 years before Brill and Noether.
Eventually one realizes Roch's version of Riemann's matrix represents the map H0(O(D))-->H1(O), induced by the sheaf sequence:
0-->O-->O(D)-->O(D)|D-->0. In particular the ancients understood and used the sheaf cohomology group H1(O) = H1(C)/H0(K).
The proof of Riemann's theorem for plane curves, although not algebraic, seems not to depend on Dirichlet's principle, since the relevant existence proof follows by writing down rational differentials. Hence later contributions of Brill - Noether and Dedekind - Weber apparently algebraicize, rather than substantiate, his results.
Riemann's philosophy that a meromorphic function is a global object, associated with its maximal domain, and determined in any subregion, "explains" why the analytic continuation of the zeta function and the Riemann hypothesis help understand primes. I.e. Euler's product formula shows the sequence of primes determines the zeta function, and such functions are understood by their zeroes and poles, so the location of zeroes must be intimately connected with the distribution of primes!
More precisely, in VII Riemann says Gauss's logarithmic integral Li(x) actually approximates the number π(x) of primes less than x, plus 1/2 the number of prime squares, plus 1/3 the number of prime cubes, etc..., hence over - estimates π(x). He inverts this relation, obtaining a series of terms Li(x^[1/n]) as a better approximation to π(x), whose proof apparently requires settling the famous "hypothesis".
In XII, Riemann both defines integrable functions, and characterizes them as functions whose points of oscillation at least e > 0, have content zero. I thought this fact depended on measure theory, but it appears rather that measure theory started here, [cf. Watson in Baker’s bilbiography].
In XIII, Riemann observes that in physics one should not expect large scale metric relations to hold in the infinitesimally small, a lesson I thought taught by physicists writing 50 years later. Elsewhere he hypothesizes that electrical impulses move at the speed of light, another assumption often credited to early 20th century physicists.
In VI, he proves a maximal set of non bounding curves has constant cardinality by the “Steinitz' exchange” method, 14 years before Steinitz' birth.
The translator apologizes for Weber’s inclusion of paper XIX on differentiation of order v where v is any real or complex number, written when Riemann was only 21, but I found it interesting: i.e. Cauchy’s theorem shows that differentiation of order v can be expressed as an integral of a vth power, which makes sense for any v, once one has the Gamma function to provide the appropriate constant multiple.
I hope this sampling from this wonderful book persuades you to read it for your own pleasure.
