Riemann hypothesis

For the musical term, see Riemannian theory.

The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first nontrivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.

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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. Many consider it to be the most important unsolved problem in pure mathematics.^{1859), after whom it is named.}

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 name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

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 nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:

The real part of every nontrivial zero of the Riemann zeta function is 1/2.

Thus, if the hypothesis is correct, all the nontrivial zeros 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.

Contents

• 1 Riemann zeta function
• 2 Origin
• 3 Consequences
  • 3.1 Distribution of prime numbers
  • 3.2 Growth of arithmetic functions
  • 3.3 Lindelöf hypothesis and growth of the zeta function
  • 3.4 Large prime gap conjecture
  • 3.5 Analytic criteria equivalent to the Riemann hypothesis
  • 3.6 Consequences of the generalized Riemann hypothesis
  • 3.7 Excluded middle
    • 3.7.1 Littlewood's theorem
    • 3.7.2 Gauss's class number conjecture
    • 3.7.3 Growth of Euler's totient
• 4 Generalizations and analogs
  • 4.1 Dirichlet L-series and other number fields
  • 4.2 Function fields and zeta functions of varieties over finite fields
  • 4.3 Arithmetic zeta functions of arithmetic schemes and their L-factors
  • 4.4 Selberg zeta functions
  • 4.5 Ihara zeta functions
  • 4.6 Montgomery's pair correlation conjecture
  • 4.7 Other zeta functions
• 5 Attempted proofs
  • 5.1 Operator theory
  • 5.2 Lee–Yang theorem
  • 5.3 Turán's result
  • 5.4 Noncommutative geometry
  • 5.5 Hilbert spaces of entire functions
  • 5.6 Quasicrystals
  • 5.7 Arithmetic zeta functions of models of elliptic curves over number fields
  • 5.8 Multiple zeta functions
• 6 Location of the zeros
  • 6.1 Number of zeros
  • 6.2 Theorem of Hadamard and de la Vallée-Poussin
  • 6.3 Zero-free regions
• 7 Zeros on the critical line
  • 7.1 Hardy–Littlewood conjectures
  • 7.2 Selberg's zeta function conjecture
  • 7.3 Numerical calculations
  • 7.4 Gram points
• 8 Arguments for and against the Riemann hypothesis
• 9 Notes
• 10 References
  • 10.1 Popular expositions
• 11 External links