Almost all people from number theory and many more from other branches of mathematics know the Riemann Hypothesis and its importance. However I am having trouble explaining why the Riemann Hypothesis is considered to one of the most important open problems in mathematics today.
What is the best way to explain the Riemann Hypothesis to a layman?
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One may interpret the Riemann Hypothesis by saying that the primes are distributed as regularly as possible: for any real number $x$ the number of prime numbers less than $x$ is approximately $Li(x)$ and this approximation is essentially square root accurate. More precisely, $$ \pi(x)=Li(x)+O(\sqrt{x}\log(x)). $$
We all know what prime numbers are. Euclid has proven that there are infinitely many of them. Experience has taught us that they get more rare when we come to ever higher numbers. Of course mathematicians want to describe the "statistics" of the primes in more precise terms. The prime number theorem (proven at the end of the $19^{\rm th}$ century) tells us that there are about $n/\log n$ primes $\leq n$ when $n$ is large. The next question then is: How much can the true number $\pi(n)$ deviate from this estimate? Working on this problem already Riemann has remarked that there is a huge technical stumbling block, nowadays called the "Riemann Hypothesis". So far nobody has managed to move this block away. Therefore mathematicians go around it: Many papers have a proviso in their introduction: "Assuming that the Riemann Hypothesis is true, we prove the following: $\> \ldots\> $".