The Riemann Zeta Function

Definition

is the original definition for this function. Using the Fundamental Theorem of Arithmetic, Euler proved that . Therefore, due to this function, we have a relationship between the integers and the primes. Certainly, this can put a very strict lower bound on the nth prime. However, this is just the beginning.

Primes

If we take the logarithm of the zeta function, we get an integral which contains pi(x), which is the number of primes not exceeding x. This formula is . This comes from the Euler product definition of the zeta function and performing a few manipulations. Also the pi function can be approximated very well with the zeta function with the following equation: .

Convergence

If s>1, then the zeta function will converge. This follows directly from the integral test. So we have a zeta function for s>1. When s=1, we have the diverging harmonic series.

Analytic Continuation

What if we need zeta(s) for s<1? Georg Friedrich Bernhard Riemann proved the following formula as an analytic continuation for zeta(s) for all . This formula is . However, it is not so easy to get arbitrary gamma values or arbitrary zeta functions, even when the infinite summation does hold. However, gamma values are known for all integers or half-integers. Similarly, zeta values are known for all positive even integers. This is equivalent to the integral analytic continuation, , where .

The Crux of the Matter: The Riemann Hypothesis

Riemann conjectured that all nontrivial zeroes of zeta(s) have real part equal to ½. In symbolic parlance, this states that if , then . This is a remarkable conjecture. Modern mathematicians have no idea how Riemann came to this conclusion, considering he did not have any computer algebra system to help him back in 1859. However, this conjecture is supported by numerical evidence. Computers have checked that the first billion zeroes (i.e. those closest to the real axis) do indeed lie on the line. With Mathematica 4.0, it is easy to generate a graph of : . Actually the graph represents the absolute value so that it is simpler to visualize. Mathematica cannot generate a complex-valued complex functional graph.

It is also possible to graph , for example: . This one never reaches 0. Clearly numerical evidence supports the Riemann hypothesis, but that is certainly not a proof.

Mathematica also allows us to create a 3D graph of : . Here is a zoom-in of the 3D graph close to the critical line: .