Grand Riemann hypothesis

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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 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[edit]

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

  • 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