nLab
Riemann hypothesis

Contents

Idea

The Riemann hypothesis or Riemann conjecture is the famous unproved statement that all nontrivial zeros of the Riemann zeta function are on the vertical line Re(z)=1/2Re(z)=1/2 in the complex plane.

Analogues of the Riemann hypothesis can be considered for many analogues of zeta functions and L-functions, here one speaks of generalized Riemann hypotheses.

An important special case over finite fields is called the Riemann–Weil conjecture and was proved by Deligne building on earlier ideas of Weil and Grothendieck. Grothendieck however expected a more natural proof using the (hypothetical) theory of motives.

References

The suggestion that the Riemann hypothesis might have a proof that is an analogue of Weil’s proof for arithmetic curves over finite fields 𝔽 q\mathbb{F}_q but generalized to the field with one element is due to

  • Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa) Asterisque, (228):4, 121-163, 1995. Columbia University Number Theory Seminar.

  • Wikipedia, Generalized Riemann hypothesis.

Revised on September 2, 2014 06:02:48 by Urs Schreiber (185.37.147.19)