A Statement of the Riemann Hypothesis

The Riemann Hypothesis, a longstanding unsolved conjecture in analytic number theory, is considered by many mathematicians to be one of the most important unsolved problems in theoretical mathematics. To understand the statement takes only a typical undergraduate mathematics education, but to find a proof would be the capstone of a mathematical career.

The Riemann zeta function is most simply described as a series, the infinite sum over n of 1/nz for complex z. This series converges when the real part of z is greater than 1, and diverges when it is less than 1. The convergence properties of the series are less straightforward when Re(z) = 1. Notice, for instance, that substituting z = 1 describes the harmonic series, which diverges, so we see that the series diverges there.

Looking at the zeta function for values of z with real part less than 1 is a bit trickier. Through an analytic technique similar in flavor to integration by parts, a representation can be obtained which is good for Re(z) > 0. From there, it is possible to use a functional equation which relates values of the function with corresponding values reflected across the line Re(z) = 1/2. It turns out that the Riemann zeta function is meromorphic on the entire complex plane with a single simple pole at the point z = 1. For those without background in complex analysis, this means that the function is infinitely differentiable and has a convergent Taylor series at every point except the pole, and if you divide by (z-1), you get a function which has this property on the entire plane.

So the zeta function is, in some sense, a very nice function. In fact, there are deep connections between the zeta function and prime number theory which require some study to appreciate. Again and again, a fundamental question emerges about the function: Where on the complex plane does the function take value zero? It is possible to show that the function is strictly non-zero on the half-plane Re(z) > 1. Using the functional equation, this also shows that the function is non-zero in Re(z) < 0, except at the negative even integers z = -2, -4, -6, ... . The real question concerns zeros that occur in what is known as the "critical strip" 0 ≤ Re(z) ≤ 1. It is known that there are infinitely many of these "non-trivial" zeros, and that they have reflective symmetry across the real axis, and across the line Re(z) = 1/2.

The Riemann hypothesis claims that the non-trivial zeros of the zeta function all lie on the line Re(z) = 1/2. The conjecture, proposed by Bernhard Riemann in 1859, has been attacked by many of the most prominent modern mathematicians, but no proof has been forthcoming. However, no disproof has emerged either. Computers have shown that the first 10 trillion zeros in the critical strip, arranged in order of positive imaginary part, satisfy the hypothesis. It is commonly accepted in the mathematical community that the hypothesis is true (hence the name “hypothesis” rather than “conjecture”), but rigorous mathematicians will not be truly satisfied until a proof has been found.

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