29 October 2004

What IS the Riemann Hypothesis?

George E. Hrabovsky, President, MAST

Where We Have Been

Last time I explained how to develop a math library. I also explained that you can't be a mathematician without doing mathematics. I also described how to learn mathematics as you do it, and how to develop projects. I then explained that I was going to delve into an area of mathematics that I knew very little about, the Riemann hypothesis.

Session 1: Where to begin?

One of the first places I go when I am unsure of a topic in mathematics is the "Encyclopedic Dictionary of Mathematics."  I will name this as source [1]: Kiyosi Itô (1993), "Encyclopedic Dictionary of Mathematics," MIT Press (Paperback Second Edition). I looked up Riemann Hypothesis in the index and came up with 450.A, I, P, Q.  This meant it was in article number 450, sections A, I, P, and Q. I began a project notebook (a bound sketchbook).

Article 450 is entitled, "Zeta Functions."  In section A, "Introduction," I discovered that there have been many invented since the 19th century, including one by George Friedrich Bernard Riemann, one of the most famous mathematicians in history. Unfortunately, the index was wrong, for there is no mention of the Riemann hypothesis in section A.

Suspecting the problem, I inspected section B, "The Riemann ."  I discovered a statement of the hypothesis.  Riemann conjectured that all nontrivial zeros of the Riemann can be found on a line described by the formula,

where,

This is the Riemann Hypothesis; we will call it RH.

Section I, "The Riemann Hypothesis," covers a great deal of detail that I do not fully understand. Some facts did come out though. The great mathematician David Hilbert showed that the formula,

where,

is equivalent to RH.

Niels Fabian Helge von Koch showed that the formula,

where,

and

is also equivalent to RH.

Section P, "Congruence of Zeta Functions of Algebraic Function Fields of One Variable or of Algebraic Curves," yielded yet another equivalence in the formula,

Section Q, "Zeta Functions of Algebraic Varieties over Finite Fields," gave a final equivalence in the form of,

where,

this is a polynomial with integer coefficients where,

and

Now I know what RH is, though I don't understand much of it at this stage. I do have the ability to find out, and I must now go through and ask a set of detailed questions for further pursuit.

What questions does this raise?

What, exactly, is a function?

What is a zeta function?

What is the Riemann zeta function?

What is the zero of a function?

What is a nontrivial zero?

What is meant by

What does mean?

What does mean?

What does O mean?

What does mean?

What does mean?

What does mean?

What does mean?

What is a congruence?

What is an algebraic function?

What is a function field?

What is an algebraic curve?

What does mean?

What does mean?

What does mean?

What does mean?

What is a variety?

What is an algebraic variety?

What is a finite field?

Book Review: "The Encyclopedic Dictionary of Mathematics"

Kiyosi Itô (1993), "Encyclopedic Dictionary of Mathematics," MIT Press (Paperback Second Edition).

This is a two-volume set of reasonably large books. The level of difficulty ranges from the elementary college level to articles that are challenging for postgraduates. The subject matter is just as varied and is indexed by subject, topic and author. The articles are cross-referenced to a high degree. While it is an encyclopedia, no effort has been made to prove the majority of theorems, nor would there be sufficient space for this.

Consider the article "Zeta Functions" as an example. This article is 26, two-column, pages long and is fully cross-referenced within sections to both additional articles and topics found in the index. If you were to work your way through this article and look up each cross-reference, you would be able to effectively develop your own minicourse in Zeta functions.

I highly recommend this reference work to anyone who is serious about doing work in mathematics of theoretical science.

Created by Mathematica  (October 19, 2004).