Riemann Hypothesis

Riemann Hypothesis without question is the most important unsolved problem in all of mathematics. The hypothesis put forth by Bernhard Riemann in 1859, states that all non- trivial zeroes of the Zeta function have a real part of one- half. Another way to put this is that the distribution of numbers with an odd number of prime factors versus those with and even numbers of prime factors is roughly 50/50.

Riemann proposes that some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern; however the German mathematician observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation lie on a certain vertical straight line. This has been checked for the first 1,500,000,000 solutions.

Experts Quotes on Riemann

This is what experts had to say about the Riemann Hypothesis. P. Sarnak said that the consequences of the Riemann Hypothesis are fantastic, the distribution of primes, and these elementary objects of arithmetic and to have the tools to study the distribution of objects. S. Gonek quoted, “The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don’t understand about the link between addition and multiplication”.

Arguments for and against the Riemann Hypothesis

Several analogues of the Riemann Hypothesis have already been proved. The proof of the Riemann Hypothesis for varieties over finite fields by Deligne is possible the single strongest theoretical reasons in favor of the Riemann Hypothesis. This provides some evidence for the more general conjecture that all zeta functions associated with automorphic forms satisfy a Riemann hypothesis.

The numerical verification is that many zeros lie on the line seems at first sight to be strong evidence for it. However analytic number theory has had many conjectures supported by large amounts of numerical evidence that turn out to be false. The problem is that the behavior is often influenced by slowly increasing functions that tend to infinity which cannot be detected by computation. Such functions occur in the theory of the zeta function controlling the behavior of its zeroes.

Questions:

• What is Riemann Hypothesis?
• What are the arguments put forth for and against the Riemann hypothesis?