The Riemann zeta function can also be defined in terms of by

Saying that the real part of any non-trivial zero is 0.5 means that when you put complex number values into the function that the real part of that value is always 0.5 if the number you get out of the function is zero.

The Riemann hypothesis is so famous because no one has been able to solve it for 150 years.

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 .

The Riemann zeta function satisfies the

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.

Naturally, the Riemann Hypothesis was one of these problems.

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 :

Itallows to generalize the Riemann hypothesis to the reals.

All of the zeros Riemann was able to calculate lay on a vertical line, and he hypothesised that of the zeta function's (nontrivial) zeros lie on this "critical line".

and "Riemann Hypothesis." From --A Wolfram Web Resource.

Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. II." 39, 681-688, 1982.

Conrey, J. B. "The Riemann Hypothesis." 50,341-353, 2003. .

In the Season 1 episode "" (2005) of the television crime drama , math genius Charlie Eppes realizes that character Ethan's daughter has been kidnapped because he is close to solving the Riemann hypothesis, which allegedly would allow the perpetrators to break essentially all internet security.

Derbyshire, J. New York: Penguin, 2004.

The Riemann Zeta function has some zeros that are easy to find which are of little interest but there are some other ones that are harder to find which is why the are called non-trivial.

Keiper, J. "The Zeta Function of Riemann." 4,5-7, 1995.

pdf A. Kovács, N Tihanyi, Efficient computing of n-dimensional simultaneous Diophantine approximation problems, Acta Univ. Sapientia Informatica, 5, 1 (2013) 16–34 pdf N. Tihanyi, Fast method for locating peak values of the Riemann-zeta function on the critical line, IEEE publication on […]