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 $\frac{1}{2} + it$ 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.{{Cite book|last1=Sarnak |first1=Peter |author1-link=Peter Sarnak |editor1-last=Arthur |editor1-first=James |editor1-link=James Arthur (mathematician) |editor2-last=Ellwood |editor2-first=David |editor3-last=Kottwitz |editor3-first=Robert |editor3-link=Robert Kottwitz |year=2005 |title=Harmonic Analysis, The Trace Formula, and Shimura Varieties |chapter=Notes on the Generalized Ramanujan Conjectures |publisher=Clay Mathematics Institute. Clay Mathematics Proceedings |volume=4 |pages=659{{ndash}}685 |location=Princeton |language=English |issn=1534-6455 |oclc=637721920 |isbn=0-8218-3844-X |chapter-url=http://web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf |access-date=November 11, 2020|url-status=live |archive-url=https://web.archive.org/web/20151004063221/http://web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf |archive-date=October 4, 2015}}
The Siegel zero, conjectured not to exist,{{Cite journal|last1=Conrey|first1=Brian|author1-link=Brian Conrey|last2=Iwaniec|first2=Henryk|author2-link=Henryk Iwaniec|date=2002|title=Spacing of zeros of Hecke L-functions and the class number problem|journal=Acta Arithmetica|language=en|volume=103|issue=3|pages=259–312|doi=10.4064/aa103-3-5|bibcode=2002AcAri.103..259C|issn=0065-1036|quote=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.|doi-access=free|arxiv=math/0111012}} 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.

grand Riemann hypothesis

Notes

References

Further reading