All the known nontrivial zeros of the Riemann zeta function have

constructed a natural space of invariant functions on the upper half plane which has eigenvalues under the Laplacian operator corresponding to zeros of the Riemann zeta function, and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space the Riemann hypothesis would follow. discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same Laplacian operator.

The Riemann zeta function, for all complex numbers  with real part , is given by the integral

Locations of the increasing peak values of the integral of the absolute value of the Riemann zeta function between successive zeros on the critical line. This can also be defined in terms of the function; if and are successive zeros of a renormalized function, , then take the integral between and of . For each successively higher value of this integral, the corresponding term of the integer sequence is rounded to the nearest integer.

to imaginary part of -th zero of Riemann zeta function.

The Riemann zeta function for negative  is 0 (those are the  of the Riemann zeta function).

introduced global zeta-functions of (quadratic) and conjectured an analogue of the Riemann hypothesis for them, which has been proven by Hasse in the genus 1 case and by in general. For instance, the fact that the , of the quadratic character of a of size q (with q odd), has absolute value

The Riemann zeta function is an important function in mathematics

Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper , but as it was not essential to his central purpose in that paper, he did not attempt a proof. Riemann knew that the non-trival zeros of the zeta function were symmetrically distributed about the line z=1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0z)

Statement In terms of zeros of the Riemann zeta-function

The Riemann hypothesis discusses zeros outside the region of convergence of this series, so it needs to be to all complex s. This can be done by expressing it in terms of the as follows. If s has positive real part, then the zeta function satisfies

All the nontrivial zeros of the Riemann zeta-function have real part

The traditional formulation of the Riemann hypothesis obscures somewhat the true importance of the conjecture. The zeta function has a deep connection to the distribution of and proved in that the Riemann hypothesis is equivalent to the following considerable strengthening of the :

Riemann zeta function - Wikipedia, the free encyclopedia

The zeros of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.

The Riemann zeta function for negative even integers is 0 ..

The Riemann hypothesis can be generalized in various ways by replacing the Riemann zeta function by the formally similar global . None of these generalizations has been proven or disproven. See .