Grand Riemann hypothesis explained

In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L-functions lie on the critical line

(1)/(2) + it with t a real number variable and i the imaginary unit.

The modified grand Riemann hypothesis is the assertion that the nontrivial zeros of all automorphic L-functions lie on the critical line or the real line.

Notes

• Robert Langlands, in his general functoriality conjectures, asserts that all global L-functions should be automorphic.^[1]
• The Siegel zero, conjectured not to exist,^[2] is a possible real zero of a Dirichlet L-series, rather near s = 1.
• L-functions of Maass cusp forms can have trivial zeros which are off the real line.

Notes and References

1. Book: Sarnak . Peter . Peter Sarnak . Arthur . James . James Arthur (mathematician) . Ellwood . David . Kottwitz . Robert . Robert Kottwitz . 2005 . Harmonic Analysis, The Trace Formula, and Shimura Varieties . Notes on the Generalized Ramanujan Conjectures . Clay Mathematics Institute. Clay Mathematics Proceedings . 4 . 659685 . Princeton . English . 1534-6455 . 637721920 . 0-8218-3844-X . http://web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf . November 11, 2020. live . https://web.archive.org/web/20151004063221/http://web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf . October 4, 2015.
2. Conrey. Brian. Brian Conrey. Iwaniec. Henryk. Henryk Iwaniec. 2002. Spacing of zeros of Hecke L-functions and the class number problem. Acta Arithmetica. en. 103. 3. 259–312. 10.4064/aa103-3-5. 2002AcAri.103..259C. 0065-1036. Conrey and Iwaniec show that sufficiently many small gaps between zeros of the Riemann zeta function would imply the non-existence of Landau–Siegel zeros.. free. math/0111012.