- Source: 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
1
2
+
i
t
{\displaystyle {\frac {1}{2}}+it}
with
t
{\displaystyle t}
a real number variable and
i
{\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.
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
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
Kata Kunci Pencarian:
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