mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the grand Riemann hypothesis is a generalisation of the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...

and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic ''L''-functions lie on the critical line

\frac + it

with

t

a real number variable and

i

the

imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...

. 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 In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

Notes

Robert Langlands Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...

, 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.

Notes

References

Further reading