What is the extended Riemann hypothesis?

The extended Riemann hypothesis is a generalization of the original Riemann hypothesis, formulated specifically for Dedekind zeta-functions associated with number fields. It conjectures that the non-trivial zeros of these global L-functions all lie on a critical line with a real part of one-half. While many mathematicians believe these conjectures are true, proven cases so far only exist for the algebraic function field case rather than the number field case.

What we know

• When the Riemann hypothesis is formulated for Dedekind zeta-functions associated with number fields, it is known as the extended Riemann hypothesis. [1, 2]
• Dedekind zeta-functions are global L-functions that are formally similar to the Riemann zeta-function. [2, 3]
• The only proven cases of generalizations of the Riemann hypothesis occur in the algebraic function field case rather than the number field case. [2, 3]

Where to go next

• how it workedWhat is the Riemann zeta function?
• the bigger pictureHow does the grand Riemann hypothesis expand further?
• a case in pointWhat are Dirichlet L-functions?
• compared withHow do function fields compare to number fields in these conjectures?
• whoWho is Atle Selberg?
• an unexpected connectionWho claimed to prove the generalized Riemann hypothesis?Louis de Branges de Bourcia, best known for proving the Bieberbach conjecture, also claimed to have proved the generalized Riemann hypothesis.

Sources

• [1]Generalized Riemann hypothesis (Wikidata Q685140) · CC0 1.0
• [2]Generalized Riemann hypothesis · CC BY-SA 4.0
• [3]Riemann hypothesis · CC BY-SA 4.0
• [4]Grand Riemann hypothesis · CC BY-SA 4.0
• [5]Louis de Branges de Bourcia · CC BY-SA 4.0