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 with a real number variable and the imaginary unit.
with 
a real number variable and 
the 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.
- The Siegel zero, conjectured not to exist, 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
- 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.
- 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
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