# Riemann Hypothesis effect on the world

What would be the effects on the world and to the individual or individuals if the Riemann Hypothesis was solved?

HallsofIvy
Homework Helper
It would mean that a number of theorems about prime numbers that have been stated "If the Riemann Hypothesis is true" would be correct.l

shmoe
Homework Helper
For those who solve it, \$1 million us and instant celebrity status in the mathematical community as well as some degree of fame outside it.

Don't expect to see any world shattering changes. It will be a huge result for Number Theorists, but the actual impact on day-to-day life of your average person will be essentially nil. Most importantly, cryptography (e-commerce, etc) will go on.

Ok, what exactly would it take to solve it? Does it simply have to produce primes one after another, in order, or is there more to it?

shmoe
Homework Helper
It doesn't "produce primes". It's a statement about the location of the zeros the Riemann Zeta function. Specifically it says that all the non-trivial zeros (the trivial zeros are at s=-2, -4, -6, ...) lie on what's known as the critical line, that is they have real part equal to 1/2. If you can prove all the non-trivial zeros satisfy this, you're done. There are many statements equivalent to the Riemann Hypothesis, like the magnitude of error term in the prime number theorem (this is why it's so important). However it seems unlikely that RH would be proven by showing this error term is small enough, the best bounds on the error term have always come from the best known zero free region of Zeta, not vice versa.

Much computational work has been done to verify the Riemann Hypothesis in finite regions, showing billions and billions of zeros satisfy RH. These types of calculations will never be able to prove RH, but would eventually disprove it if it were false.

I have no idea what you currently know and this is a very deep subject. If you want more details I'm happy to do my best to answer specific questions. You might try searching around this forum, this sort of question has come up before. There have also been a few recent books aimed at the general public on zeta (the clay prize gave it a higher profile), you might want to check them out.

arildno