I've never heard of a Riemann sequence, but the zeta function isn't too difficult a topic.
The zeta function, written
In other words, it's the sum of the reciprocals of the s-th powers of the integers.
In this form, it only makes sense for numbers with real part greater than 1. You might have heard about how the reciprocals of the integers, the harmonic series, adds to infinity.
It is possible to make the zeta function meaningful for all complex numbers other than 1 via what is called "analytic continuation". We eventually end up with some complicated looking thing that we don't need to get into the details of.
There are certain numbers s for which the zeta function gives zero; these numbers are, somewhat confusingly, called "zeros" of the zeta function. There are the trivial zeros, which are the negative even integers. Don't ask why they work even though it seems they shouldn't; it's a consequence of the analytic continuation.
Then there are the non-trivial zeros. The Riemann Hypothesis states that these zeros all have real part 1/2. All of the non-trivial zeros found so far have real part 1/2, but that doesn't mean that they all do.
If the Riemann Hypothesis proves correct, then we'd end up with a very good bound on the distribution of primes.
There's also a hope that the Riemann Hypothesis will be generalizable to a larger class of functions called L-functions, which are like the zeta function but with different numerators instead of 1. If the generalized Riemann Hypothesis is true then we would get a lot of information about deep number-theoretic and algebraic structures.
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