Riemann hypothesis

Critical line theoremcritical line1stBeurling–Nyman criterionGeneralized Riemann HypothesisRiemann zerosRiemann's hypothesisThe Riemann Hypothesistrivial zeros
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1⁄2.wikipedia
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Conjecture

conjecturalconjecturesconjectured
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1⁄2.
Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

Hilbert's problems

Hilbert problems23 problems23 unsolved problems
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, comprise Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems.
Some of them are propounded precisely enough to enable a clear affirmative or negative answer, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis).

Millennium Prize Problems

Millennium ProblemsMillennium PrizeMillennium problem
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, comprise Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems.
The problems are the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Poincaré conjecture, Riemann hypothesis, and Yang–Mills existence and mass gap.

Prime number

primeprime factorprime numbers
It is of great interest in number theory because it implies results about the distribution of prime numbers.
Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem.

Clay Mathematics Institute

Clay Research FellowClay InstituteClay Research Fellowship
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, comprise Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems.
Of the initial 23 Hilbert problems, most of which have been solved, only the Riemann hypothesis (formulated in 1859) is included in the seven Millennium Prize Problems.

Riemann zeta function

zeta functionRiemann zeta-functionRiemann's zeta function
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1⁄2.
and, according to the Riemann hypothesis, they all lie on the line

Hilbert's eighth problem

8th
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, comprise Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems.
It concerns number theory, and in particular the Riemann hypothesis, although it is also concerned with the Goldbach Conjecture.

Mertens conjecture

The Riemann hypothesis puts a rather tight bound on the growth of M, since disproved the slightly stronger Mertens conjecture
In mathematics, the Mertens conjecture is the disproven statement that the Mertens function M(n) is bounded by \sqrt{n}, which implies the Riemann hypothesis.

John Edensor Littlewood

J. E. LittlewoodLittlewoodJ.E. Littlewood
for every positive ε is equivalent to the Riemann hypothesis (J.E. Littlewood, 1912; see for instance: paragraph 14.25 in ).
One of the problems that Barnes suggested to Littlewood was to prove the Riemann hypothesis, an assignment at which he did not succeed.

Lindelöf hypothesis

The Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,
This hypothesis is implied by the Riemann hypothesis.

Redheffer matrix

(For the meaning of these symbols, see Big O notation.) The determinant of the order n Redheffer matrix is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants.
The determinants of the Redheffer matrices are immediately tied to the Riemann Hypothesis (RH) through this intimate relation with the Mertens function as the RH is equivalent to showing that for all (sufficiently small).

Jérôme Franel

Franel, JérômeJerome Franel
Another example was found by Jérôme Franel, and extended by Landau (see ).
He is mainly known through a 1924 paper, in which he establishes the equivalence of the Riemann hypothesis to a statement on the size of the discrepancy in the Farey sequences, and which is directly followed (in the same journal) by a development on the same subject by Edmund Landau.

Zeros and poles

polepoleszero
The zeta function can be extended to these values too by taking limits, giving a finite value for all values of s with positive real part except for the simple pole at s = 1.
. Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along

Chebyshev function

Chebyshev's functionChebyshev's second functionsecond Chebyshev function
where ψ(x) is Chebyshev's second function.
Furthermore, under the Riemann hypothesis,

Mertens function

M''(12)
From this we can also conclude that if the Mertens function is defined by
However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x), namely M(x) = O(x 1/2 + ε ).

On the Number of Primes Less Than a Given Magnitude

1859 paper1859 paper on the zeta-functiona single short paper
Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes (x) less than or equal to a given number x, which he published in his 1859 paper "On the Number of Primes Less Than a Given Magnitude".
*The Riemann hypothesis, that all (nontrivial) zeros of ζ(s) have real part 1/2.

Number theory

number theoristcombinatorial number theorytheory of numbers
It is of great interest in number theory because it implies results about the distribution of prime numbers.
The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy–Littlewood conjectures), the Waring problem and the Riemann hypothesis.

Möbius function

Moebius functionMobius functionμ(''n'')
where μ is the Möbius function.
and the Riemann hypothesis.

Dirichlet eta function

Landau's problem on the Dirichlet eta functionalternating zeta functionDirichlet eta-function
One begins by showing that the zeta function and the Dirichlet eta function satisfy the relation
Under the Riemann hypothesis, the zeros of the eta function would be located symmetrically with respect to the real axis on two parallel lines, and on the perpendicular half line formed by the negative real axis.

Prime-counting function

prime counting functioncounting prime numbersdenoting the number of prime numbers
Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes (x) less than or equal to a given number x, which he published in his 1859 paper "On the Number of Primes Less Than a Given Magnitude".
The Riemann hypothesis suggests that every such non-trivial zero lies along

Generalized Riemann hypothesis

extended Riemann hypothesisERHGeneralised Riemann hypothesis
The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet L-functions.
The Riemann hypothesis is one of the most important conjectures in mathematics.

Riesz function

Riesz criterionRiesz-type formula
The Riesz criterion was given by, to the effect that the bound
In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series

Li's criterion

Related is Li's criterion, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.
In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis.

Prime number theorem

distribution of primesdistribution of prime numbersprime number theorem for arithmetic progressions
In particular the error term in the prime number theorem is closely related to the position of the zeros.
, the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.

Cramér's conjecture

conjecturedCramer conjectureCramér conjecture
This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: Cramér's conjecture implies that every gap is O((log p) 2 ), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis.
:on the assumption of the Riemann hypothesis.