Source: Wikipedia. Pages: 72. Chapters: Goldbach's conjecture,
Prime number theorem, Elliptic curve, Elliptic function,
Goldbach's weak conjecture, De Bruijn-Newman constant,
Seventeen or Bust, Stirling's approximation, Riemann
hypothesis, Modular group, Prime-counting function, Modular form,
Transcendence theory, Kloosterman sum, Eisenstein series,
Hardy-Littlewood circle method, Lindelöf hypothesis, Apéry's
constant, Chebotarev's density theorem, Classical modular
curve, Smooth number, Exponential sum, Dickman function,
Schinzel's hypothesis H, Bateman-Horn conjecture, Abstract
analytic number theory, Bombieri norm, On the Number of Primes Less
Than a Given Magnitude, Linnik's theorem, Effective results in
number theory, Real analytic Eisenstein series, Character sum,
Artin's conjecture on primitive roots, Euler's four-square
identity, Cramér's conjecture, Lambert series, Fermat polygonal
number theorem, Gilbreath's conjecture, Riesel number,
Degen's eight-square identity, Montgomery's pair
correlation conjecture, Multiplicative number theory, Dirichlet
density, Chen's theorem, Elliott-Halberstam conjecture, Landau
prime ideal theorem, Odlyzko-Schönhage algorithm,
Bombieri-Vinogradov theorem, Mahler measure, Second
Hardy-Littlewood conjecture, Constant problem, Kuznetsov trace
formula, Perron's formula, Kronecker limit formula, Maier's
theorem, Friedlander-Iwaniec theorem, Birch's theorem,
Brauer-Siegel theorem, Siegel-Walfisz theorem, Hardy-Ramanujan
theorem, The Music of the Primes, Brun-Titchmarsh theorem,
Landau-Ramanujan constant, Vaughan's identity, Petersson trace
formula, Hua's lemma, Artin conjecture, Riemann-von Mangoldt
formula, Siegel G-function. Excerpt: In mathematics, the Riemann
hypothesis, proposed by Bernhard Riemann (1859), is a conjecture
about the location of the zeros of the Riemann zeta function which
states that all non-trivial zeros have real part 1/2. The name is
also used for some closely related analogues, such as the Riemann
hypothesis for curves over finite fields. The Riemann hypothesis
implies results about the distribution of prime numbers that are in
some ways as good as possible. Along with suitable generalizations,
it is considered by some mathematicians to be the most important
unresolved problem in pure mathematics (Bombieri 2000). The Riemann
hypothesis is part of Problem 8, along with the Goldbach
conjecture, in Hilbert's list of 23 unsolved problems, and is
also one of the Clay Mathematics Institute Millennium Prize
Problems. Since it was formulated, it has withstood concentrated
efforts from many outstanding mathematicians. In 1973, Pierre
Deligne proved an analogue of the Riemann Hypothesis for zeta
functions of varieties defined over finite fields. The full version
of the hypothesis remains unsolved, although modern computer
calculations have shown that the first 10 trillion zeros lie on the
critical line. The Riemann zeta function ¿(s) is defined for all
complex numbers s ¿ 1. It has zeros at the negative even integers
(i.e. at s = -2, -4, -6, ...). These are called the trivial zeros.
The Riemann hypothesis is concerned with the non-trivial zeros, and
states that: The real part of any non-trivial zero of the Riemann
zeta function is 1/2.Thus the non-trivial zeros should lie on the
critical line, 1/2 + it, where t is a real number and i is the
imaginary unit. There are several popular books on the Riemann
hypothesis, such as Derbyshire (2003), Rockmore (2005), Sabbagh
(2003), du Sautoy (2003...
