Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecturezeros 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. about the distribution of the
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 that the Riemann hypothesis held true 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 ζ(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). The books Edwards (1974), Patterson (1988) and Borwein et al. (2008) give mathematical introductions, while Titchmarsh (1986), Ivić (1985) and Karatsuba & Voronin (1992) are advanced monographs.
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