Is the Riemann hypothesis probably true?

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much.

What would happen if Riemann hypothesis false?

The Riemann hypothesis implies a bound on the error term in the prime number theorem. Specifically, it implies that π(x)=xlogx+O(√xlogx). If the Riemann hypothesis is shown not to be true, then we will not know that this result is true.

What is the Riemann hypothesis?

The Riemann hypothesis is that all nontrivial zeros are on this line. Proving the Riemann Hypothesis would allow us to greatly sharpen many number theoretical results. For example, in 1901 von Koch showed that the Riemann hypothesis is equivalent to:

What is Riemann’s equivalent statement?

An equivalent statement (Riemann’s actual statement) is that all the roots of the Riemann xi function ξ(s) are real. The Riemann Zeta Function and Prime Numbers. Using the truth of the Riemann hypothesis as a starting point, Riemann began studying its consequences.

What is the Riemann prime counting function?

Riemann used π(x) to define his own prime counting function, the Riemann prime counting function J(x). It is defined as: The first thing to notice about this function is that it is not infinite. At some term, the counting function will be zero because there are no primes for x < 2.

What is the Riemann zeta function for negative integers?

However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros. The Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that: The real part of every non-trivial zero of the Riemann zeta function is 12.