Grand Riemann hypothesis
Appearance
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 with a real number variable and 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.[citation needed]
- 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
- ^ 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, vol. 27, Springer-Verlag, ISBN 0-387-72125-8