Generalized Riemann hypothesis - Wikipedia

 proved that the Riemann Hypothesis is true if and only if the space of functions of the form

Riemann hypothesis - Simple English Wikipedia, the …

The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that

showed that the Riemann hypothesis is true if and only if the integral equation

Riemann Hypothesis - Wikipedia | Conjecture | Analysis

The practical uses of the Riemann hypothesis include many propositions which are known to be true under the Riemann hypothesis, and some which can be shown to be equivalent to the Riemann hypothesis.

proved that the Riemann hypothesis is equivalent to the

The Riemann hypothesis is part of , along with the , in 's list of , and is also one of the . Since it was formulated, it has withstood concentrated efforts from many outstanding mathematicians. In 1973, proved that the Riemann hypothesis held true over finite fields. The full version of the hypothesis remains unsolved, although modern computer calculations have shown that the first 10 trillion zeros lie on the critical line.

Riemann hypothesis - The Full Wiki

For an example from group theory, if g(n) is given by the maximal order of elements of the Sn of degree n, then showed that the Riemann hypothesis is equivalent to the bound

The Riemann Hypothesis For Dummies @ Things Of Interest

In mathematics, the Riemann hypothesis, proposed by (), is a about the distribution of the of the which states that all zeros of the Riemann have real part 1/2. The name is also used for some closely related analogues, such as the .

In theorem 1.2 for the relation of Riemann's Hypothesis with ..

The Riemann hypothesis has various weaker consequences as well; one is the on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,

Riemann Hypothesis/Biography of Riemann - Wikibooks

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 ().

Riemann hypothesis from Wikipedia and other ..

There has been considerable excitement about these connections between the Riemann Hypothesis and quantum mechanics, but although they have inspired several new lines of attack, the problem still continues to resist. We don't really know why RMT methods work in calculating the moments of the Riemann zeta function. However, they have been used to suggest answers to some other long-standing and important problems relating to the zeta function.

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Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in solving it. Some typical examples are as follows.