Riemann hypothesis



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/L-functions. An important case over the 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.


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.

Revised on July 21, 2014 11:21:03 by Urs Schreiber (