Generalised Riemann hypothesis

It would also be nice to include consequences of the generalized Riemann hypothesis ..
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Generalized Riemann hypothesis - WikiVisually

I would be nice if someone could prove "the generalized Riemann hypothesis". Or maybe prove it be unprovable or something.
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assuming the generalized Riemann hypothesis for the zeros of the Dirichlet -functions
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to PH assuming the Generalized Riemann Hypothesis ..

Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts). of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros. This is because the Dedekind zeta functions factorize as a product of powers of , so zeros of Artin L-functions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some : these can have multiple zeros at the real point of their critical line; the predicts that the multiplicity of this zero is the rank of the elliptic curve.

Most such results, likewise, remain unproven without assuming the Riemann hypothesis
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In addition to simpler proofs of existing theorems, new theorems by Landauinclude important facts about Riemann's Hypothesis;facts about Dirichlet series;key lemmas of analysis;a result in Waring's Problem;a generalization of the Little Picard Theorem;a partial proof of Gauss' conjecture about the density of classesof composite numbers;and key results in the theory of pecking orders, e.g.

Results 1 - 10 of about 53,100 for "assuming the generalized Riemann hypothesis".
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List of the Greatest Mathematicians ever and their Contributions

The prime number theorem implies that on average, the between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. proved that, assuming the Riemann hypothesis, every gap is O(√p log p). This is a case when even the best bound that can currently be proved using the Riemann hypothesis is far weaker than what seems to be true: implies that every gap is O(log(p)2) which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture ().

The 8000th Busy Beaver number eludes ZF set theory: …

The Riemann hypothesis implies results about the distribution of that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in ().

Einstein: "Relativity and the Problem of Space"

We show that one can stillobtain some weaker asymptotic results assuming the Generalized RiemannHypothesis (GRH) in place of the twin prime conjecture.

A Multiresolution Triangular Plate-Bending Element Method

:We show that the problem of derandomizing Noether's Normalization Lemma (NNL) that lies at the heart of the wild problem of classifying tuples of matrices can be brought down from , where it was earlier, to unconditionally, to assuming the Generalized Riemann Hypothesis (GRH), and even further to assuming the black-box derandomization hypothesis for symbolic trace (or equivalently determinant) identity testing. Furthermore, we show that the problem of derandomizing Noether's Normalization Lemma for any explicit variety can be brought down from , where it is currently, to assuming a strengthened form of the black-box derandomization hypothesis for polynomial identity testing (PIT). These and related results reveal that the fundamental problems of Geometry (classification) and Complexity Theory (lower bounds and derandomization) share a common root difficulty, namely, the problem of overcoming the formidable vs. gap in the complexity of NNL for explicit varieties. We call this gap the . On the positive side, we show that NNL for the ring of invariants for any finite dimensional rational representation of the special linear group of fixed dimension can be brought down from to quasi- unconditionally.

Authors: YiMing Xia Comments: 17 Pages

The Riemann hypothesis can be generalized by replacing the Riemann zeta-function by the formally similar, but much more general, global . In this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta-function, which accounts for the true importance of the Riemann hypothesis in mathematics.