Grand Riemann hypothesis

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 ${\displaystyle {\frac {1}{2}}+it}$ with ${\displaystyle t}$ a real number variable and ${\displaystyle i}$ 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.

References

1. Sarnak, Peter (2005). "Notes on the Generalized Ramanujan Conjectures" (PDF). In Arthur, James; Ellwood, David; Kottwitz, Robert (eds.). Harmonic Analysis, The Trace Formula, and Shimura Varieties. Vol. 4. Princeton: Clay Mathematics Institute. Clay Mathematics Proceedings. pp. 659–685. ISBN 0-8218-3844-X. ISSN 1534-6455. OCLC 637721920. Archived (PDF) from the original on October 4, 2015. Retrieved November 11, 2020.
2. Conrey, Brian; Iwaniec, Henryk (2002). "Spacing of zeros of Hecke L-functions and the class number problem". Acta Arithmetica. 103 (3): 259–312. arXiv:math/0111012. Bibcode:2002AcAri.103..259C. doi:10.4064/aa103-3-5. ISSN 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.

Further reading

• Borwein, Peter B. (2008), The Riemann hypothesis: a resource for the aficionado and virtuoso alike, CMS books in mathematics, vol. 27, Springer-Verlag, ISBN 978-0-387-72125-5