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 ${\displaystyle {\frac {1}{2}}+it}$ with ${\displaystyle t}$ a real number variable and ${\displaystyle i}$ 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

• It is widely believed that all global L-functions are automorphic L-functions.
• The Siegel zero, conjectured not to exist,[1] 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

1. Conrey, B.; Iwaniec, H. (2002). "Spacing of zeros of Hecke L-functions and the class number problem". Acta Arithmetica. 103 (3): 259–312. 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, 27, Springer-Verlag, ISBN 0-387-72125-8