Riemann hypothesis

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. It is of great interest in number theory because it implies results about the distribution of prime numbers.

It was proposed by Bernhard Riemann (1859), after whom it is named.

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.

The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global L-functions. In this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2.

It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics.

The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet L-functions. In particular it implies the conjecture that Siegel zeros (zeros of L-functions between 1/2.and 1) do not exist.

The extended Riemann hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields.

The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.

The Riemann hypothesis and some of its generalizations, along with Goldbach’s conjecture and the twin prime conjecture, make up 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.

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