jacksonwalter said:What about the Riemann Zeta function makes it so difficult to prove that all the zeros have real part 1/2? Is it that we lack the discoveries and tools necessary, or we just aren't creative enough, or maybe both? Same question for Goldbach's. Fermat's seemed to rely on elliptic curves which have really only been invented/discovered rather recently in relation the time the problem was posed.
CRGreathouse said:We're making progress on the Riemann hypothesis, but there's still a lot of work to go. Similarly, Goldbach's conjecture has had great progress -- we've nearly proved the weak version (only finitely many verifications to go!).
In that sense they're unlike the Collatz conjecture where we appear to lack the tools to attack the problem.
jacksonwalter said:So what consequences will occur as a result of proving the Riemann Hypothesis?
CRGreathouse said:There are many, but the most important one to me is the tight bound on pi(x) due to Schoenfeld.
jacksonwalter said:So what exactly do you mean by us having made great progress?
The Zeta Function, denoted by ζ(s), is a mathematical function that arises in many areas of mathematics, including number theory, complex analysis, and physics. It is defined for all complex numbers s with real part greater than 1 and is given by the infinite series ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ..., where s is the variable.
The zeros of the Zeta Function are the values of s for which ζ(s) = 0. These values are known as the non-trivial zeros, as they lie in the critical strip 0 < Re(s) < 1. The only known real zero is s = 1, known as the trivial zero.
The distribution of the zeros of the Zeta Function is closely related to the distribution of prime numbers. The Riemann Hypothesis, which states that all non-trivial zeros lie on the critical line Re(s) = 1/2, is one of the most famous unsolved problems in mathematics and has important consequences in number theory.
There is no known formula for calculating the zeros of the Zeta Function. However, they can be approximated using numerical methods, such as the Riemann-Siegel formula or the Lindelöf hypothesis. The first 10 trillion zeros have been computed and verified to lie on the critical line, providing strong evidence for the Riemann Hypothesis.
The Zeta Function has many applications in mathematics and physics. In number theory, it is used to study the distribution of prime numbers and to prove results such as the Prime Number Theorem. In physics, it is used to calculate the Casimir effect and to study the behavior of quantum systems. It also has connections to other areas of mathematics, such as algebraic geometry and representation theory.