25 Facts About 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.

The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:.

Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product.

Riemann hypothesis's formula was given in terms of the related function.

Riemann hypothesis knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line and he knew that all of its non-trivial zeros must lie in the range = 1.

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Von Koch proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem.

Dudek proved that the Riemann hypothesis implies that for all there is a prime satisfying.

Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to the primes counting function above.

Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from µ.

The Riemann hypothesis is equivalent to several statements showing that the terms of the Farey sequence are fairly regular.

Riemann hypothesis implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip.

Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving it.

Nyman proved that the Riemann hypothesis is true if and only if the space of functions of the form.

Since the Riemann hypothesis is equivalent to the claim that all the zeroes of H are real, the Riemann hypothesis is equivalent to the conjecture that.

Several results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder.

Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.

Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global L-functions.

The Riemann hypothesis can be extended to the L-functions of Hecke characters of number fields.

Grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.

Hilbert and Polya suggested that one way to derive the Riemann hypothesis would be to find a self-adjoint operator, from the existence of which the statement on the real parts of the zeros of ? would follow when one applies the criterion on real eigenvalues.

Riemann hypothesis showed that this in turn would imply that the Riemann hypothesis is true.

Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, a distribution with discrete support whose Fourier transform has discrete support.

De la Vallee-Poussin proved that if is a zero of the Riemann zeta function, then for some positive constant C In other words, zeros cannot be too close to the line there is a zero-free region close to this line.

Mathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth.

Some arguments for and against the Riemann hypothesis are listed by Sarnak, Conrey, and Ivic, and include the following:.

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