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.
• 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.

Further reading

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