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- Riemann_hypothesis abstract "In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the nontrivial zeros of the Riemann zeta function all 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. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann hypothesis, along with the Goldbach conjecture, is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute Millennium Prize Problems.The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros. The Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that:The real part of every non-trivial zero of the Riemann zeta function is 1/2.Thus the non-trivial zeros should lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit.There are several nontechnical books on the Riemann hypothesis, such as Derbyshire (2003), Rockmore (2005), Sabbagh (2003),du Sautoy (2003). The books Edwards (1974), Patterson (1988), Borwein et al. (2008) and Mazur & Stein (2014) give mathematical introductions, whileTitchmarsh (1986), Ivić (1985) and Karatsuba & Voronin (1992) are advanced monographs.".
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- Riemann_hypothesis authorLink "Bernhard Riemann".
- Riemann_hypothesis authorlink "Alain Connes".
- Riemann_hypothesis authorlink "Anatolii Alexeevitch Karatsuba".
- Riemann_hypothesis authorlink "J.-P. Serre".
- Riemann_hypothesis authorlink "Pierre Deligne".
- Riemann_hypothesis authorlink "Pál Turán".
- Riemann_hypothesis first "A. A.".
- Riemann_hypothesis first "A. F.".
- Riemann_hypothesis first "Bernhard".
- Riemann_hypothesis first "Jean-Pierre".
- Riemann_hypothesis first "Pierre".
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- Riemann_hypothesis last "Connes".
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- Riemann_hypothesis last "Karatsuba".
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- Riemann_hypothesis quote """".
- Riemann_hypothesis quote ""…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation."".
- Riemann_hypothesis source "Riemann's statement of the Riemann hypothesis, from .".
- Riemann_hypothesis title "Zeta-function".
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- Riemann_hypothesis year "1859".
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- Riemann_hypothesis subject Category:1859_introductions.
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- Riemann_hypothesis subject Category:Millennium_Prize_Problems.
- Riemann_hypothesis subject Category:Unsolved_problems_in_mathematics.
- Riemann_hypothesis subject Category:Zeta_and_L-functions.
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