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The Riemann Hypothesis

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H.M. Edwards - Riemann's Zeta Function

P. Sarnak, from "Prime Time" by E. Klarreich (New Scientist, 11/11/00)

"The consequences [of the Riemann Hypothesis] are fantastic: the distribution of primes, these elementary objects of arithmetic. And to have tools to study the distribution of these of objects."

H. Iwaniec, quoted in K. Sabbagh's Dr. Riemann's Zeros (Atlantic, 2002), p.30

"If [the Riemann Hypothesis is] not true, then the world is a very different place. The whole structure of integers and prime numbers would be very different to what we could imagine. In a way, it would be more interesting if it were false, but it would be a disaster because we've built so much round assuming its truth."

P. Sarnak, quoted in K. Sabbagh's Dr. Riemann's Zeros (Atlantic, 2002), p.30

"If there are lots of zeros off the line - and there might be - the whole picture is just horrible, horrible, very ugly. It's an Occam's razor sort of thing, you either have absolutely beautiful behaviour of prime numbers, they behave just like you want them to behave, or else it's really bad."

S. Gonek, quoted in Dr. Riemann's Zeros (Atlantic, 2002), p.112

"The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don't understand about the link between addition and multiplication."

B. Conrey, quoted in Dr. Riemann's Zeros (Atlantic, 2002), p.160

"[The Riemann Hypothesis] is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication. It's a gaping hole in our understanding..."

A. Connes, quoted in Dr. Riemann's Zeros (Atlantic, 2002), p.208
 

Basic introduction to the Riemann Hypothesis (C. Caldwell)

Eric Weisstein's notes on the Riemann Hypothesis

In-depth examination of issues surrounding the Riemann Hypothesis (D. Bump)

Introduction to the Riemann Hypothesis (K. Spiliopoulos)

G. Pugh's excellent "The Riemann Hypothesis in a Nutshell", including a Z(t) plotting applet

J. Brian Conrey, "The Riemann Hypothesis", Notices of the AMS (March 2003) - a very nice, comprehensive introduction to the RH

J. Perry's introductory notes on the Riemann Hypothesis

WWN notes on the Riemann Hypothesis (part of a work-in-progress)

Riemann Hypothesis links

Wikipedia entry
 

Z. Rudnick, "Number theoretic background", Proceedings of a summer school in Bologna, August 2001

This covers all the number theory necessary for a basic understanding of the Riemann Hypothesis, which is covered in its final section.
 

Critical Strip Explorer v0.67, a wonderful applet produced by Raymond Manzoni for this site - explore the behaviour of the Riemann zeta function in and around the critical strip in a highly visual, interactive way. The resulting images are quite astonishing!
 

Riemann's original eight-page paper
PostScript, English translation     other formats

"Riemann wrote only one article on the theory of numbers, published in 1859. This paper radically redrew the landscape of the subject. The specific approach to the distribution of prime numbers he developed, both simple and revolutionary, consists of appealing to Cauchy's theory of holomorphic functions, which at that time was a relatively recent discovery."

[G. Tenenbaum and M. Mendès France, from The Prime Numbers and Their Distribution (AMS, 2000)]
 

"The Riemann Hypothesis and its generalisations" - WWN notes, part of a work-in-progress, see also the subsections:

• Generalised Riemann Hypothesis
• Extended Riemann Hypothesis
• Grand Riemann Hypothesis

J. Baez, This Week's Finds in Mathematical Physics week 217 includes very helpful discussion of the Riemann Hypothesis, Extended Riemann Hypothesis, Grand Riemann Hypothesis, Weil Conjectures, Langlands Programme, the functional equations of zeta and L-functions, modularity of theta functions, etc.
 

E. Bach, "Is the Riemann Hypothesis necessary?"
 

The Clay Mathematics Institute offers $1,000,000 for a proof of the Riemann Hypothesis

an extremely thorough mathematical description of the Riemann Hypothesis (with historical background, etc.) provided by Enrico Bombieri for the purposes of this competition

video recording of an introductory lecture by J. Vaaler on the RH (one of the Clay Foundation's "Millenium Lectures") [requires RealPlayer]

K. Sabbagh, Dr. Riemann's Zeros: The Search for the $1 Million Solution to the Greatest Problem in Mathematics (Atlantic Books, 2002) - a recently published popular account of the Riemann hypothesis, to be published in the U.S. in April as The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Farrar, Straus and Giroux)

Two more books of a similar nature followed in 2003:

J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, (JHP, 2003)

Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (HaperCollins, 2003)

Observer profile of du Sautoy

Here is K. Leutwyler's comparitive review of all three books from Scientific American.

Here is another, by D. Lim, from The Village Voice.

...and another by J.C. Alexander
 

some proposed proofs of the Riemann Hypothesis (some more serious than others...)

some reformulations of the Riemann Hypothesis

Professor J.E. Littlewood's brief argument as to why he believes the Riemann Hypothesis to be false.

Set theorist and mathematical philosopher Gregory Chaitin discusses the possibility that the RH might be undecidable, i.e., there is no proof.
 

a popular exposition on the Riemann Hypothesis which appeared in New Scientist (11/11/00)

"The Mark of Zeta" - introductory essay on RH and Riemann's zeta function (I. Peterson)

"The Return of Zeta" - sequel article by I. Peterson on links between the RH, random matrix theory and quantum chaos

K. Sabbagh, "Beautiful Mathematics", Prospect (January 2002)

B. Schechter, "143-year-old problem still has mathematicians guessing". A fairly good New York Times article on recent Zeta-functions conference at the Courant Institute (02/07/02). This online version requires a user ID and password, but registration is free and only takes a couple of minutes.
 

ZetaGrid - Verification of the Riemann Hypothesis (a project coordinated by S. Wedeniwski of IBM Deutschland)

"Today, we have better resources to verify or falsify Riemann's hypothesis. First the high-speed computers, then the networks have increased the capacity of calculations. Now we want to go one step further by bundling up the resources into a grid network. Therefore, I invite all interested people to participate in the calculation of the zeros of the Riemann zeta function for a new record."

S. Wedeniwski, "Computations connected with the verification of the Riemann Hypothesis" (useful overview with history and references)
 

A.R. Booker, "Turing and the Riemann Hypothesis", Notices of the AMS 53 (2006) 1208-1211