Borwein, J. M. and Bradley, D. M. "Searching Symbolically for Apéry-Like Formulae for Values of the Riemann Zeta Function." 30, 2-7, 1996.
Sondow, J. "Analytic Continuation of Riemann's Zeta Function and Values at Negative Integers via Euler's Transformation of Series." 120, 421-424, 1994.
Biane, P.; Pitman, J.; and Yor, M. "Probability Laws Related to the Jacobi Theta and Riemann Zeta Functions, and Brownian Excursions." 38, 435-465, 2001.
Srivastava, H. M. "Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments." 246, 331-351, 2000.
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.
Abramowitz, M. and Stegun, I. A. (Eds.). "Riemann Zeta Function and Other Sums of Reciprocal Powers." §23.2 in New York: Dover, pp. 807-808, 1972.
The inverse of the Riemann zeta function , plotted above, is the asymptotic density of th-powerfree numbers (i.e., numbers, numbers, etc.). The following table gives the number of th-powerfree numbers for several values of .
Riemann's other masterpieces include tensor analysis,the theory of functions, and a key relationship betweensome differential equation solutions and hypergeometric series.
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).
The Riemann zeta function may be computed analytically for using either or with the appropriate . An unexpected and important formula involving a product over the was first discovered by Euler in 1737,
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.
(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.
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.
Here, the sum on the right-hand side is exactly the (sometimes also called the alternating zeta function). While this formula defines for only the , equation (◇) can be used to analytically continue it to the rest of the . can also be performed using . A globally convergent series for the Riemann zeta function (which provides the of to the entire except ) is given by
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.