mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 pure ...

and

generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...

. 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 mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...

. 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 A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...

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 In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero, also known as exceptional zeroSee Iwaniec (2006).), named after Edmund Landau and Carl Ludwig Siegel, is a type of potential cou ...

, 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