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Deligne's contribution was to supply the estimate of the eigenvalues of the Frobenius endomorphism, considered the geometric analogue of the Riemann hypothesis. It also led to a proof of the Lefschetz hyperplane theorem and the old and new estimates of the classical exponential sums, among other applications.
The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.
In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann , is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by Deligne , using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.
The extended Riemann hypothesis asserts that for every number field K and every complex number s with ζ K (s) = 0: if the real part of s is between 0 and 1, then it is in fact 1/2. The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q, with ring of integers Z.
This leads to an infinite cyclic monodromy group and a covering of C \ {0} by a helicoid (an example of a Riemann surface). In mathematics , monodromy is the study of how objects from mathematical analysis , algebraic topology , algebraic geometry and differential geometry behave as they "run round" a singularity .
These results also follow from the Weil conjectures, except for the case k = 1, where it is a result of Deligne & Serre (1974). The Ramanujan–Petersson conjecture for Maass forms is still open (as of 2022) because Deligne's method, which works well in the holomorphic case, does not work in the real analytic case.
A proposed evolutionary hypothesis is that men and women evolved different mental abilities to adapt to their different roles, including labor-based roles, in society. [51] For example, "ancestral women more often foraged for fruits, vegetables, and roots over large geographic regions."