nLab Riemann hypothesis

Context

Higher algebra

higher algebra

universal algebra

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 $Re(z)=1/2$ 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.

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 $\mathbb{F}_q$ 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 (89.15.239.217)