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 + it with t a real number 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

• It is widely believed that all global L-functions are automorphic L-functions.
• 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

• Borwein, Peter B. (2008), The Riemann hypothesis: a resource for the afficionado and virtuoso alike, CMS books in mathematics, 27, Springer-Verlag, ISBN 0387721258