The Riemann Hypothesis and the Roots of the Riemann Zeta Function

The Clay Mathematics Institute has offered a prize of $1,000,000 for a resolution of the Riemann Hypothesis that all of the zeros of the Riemann zeta function that lie in the critical strip 0 < s < 1 are on the line s = 1/2. Perhaps because of the prize, recently many proofs (and at least one disproof) have been put forward by a variety of authors. Two proofs made it as far as the web site arXiv.org, but were withdrawn by their authors after flaws were pointed out.

This book purports to give a proof. Its author, a member of the American Mathematical Society, holds the Ph. D. degree in chemical engineering (1987, University of Illinois). He has worked for Eastman Kodak and Exxon Research and currently has a “wealth advisory practice” in Virginia. His book, 140 pages long, was published by BookSurge Publishing, an organization that enables on-demand publishing.

The author does not say at whom his book is aimed, but the level of mathematics is sufficiently high that I doubt that it could be read by anyone other than professional mathematicians.

As might be expected, the book contains a good deal of what could be called padding. For example, there are graphs of the first ten roots of the Riemann zeta function, six pages devoted to a chart of a sequence converging to its first imaginary root, a constant given to 1026 significant figures, and an excerpt from an encyclopedia about the Gordian knot.

For this reason, and others, I can’t say if his proof is correct. He says that the series for the zeta function for s > 1, the sum of the reciprocals of the sth powers of the positive integers, diverges everywhere in the critical strip 0 < s < 1 but that it “does, in fact, converge at the roots in the critical strip—and only at the roots in the critical roots in the critical strip—in a special geometrical sense.” What this means was not clear to me and I did not exert myself sufficiently to make it clear.

A good use for the book, I think, would be for an instructor in a course in analytic number theory to give it to a student with the assignment of seeing what, if anything, is there. If the proof is valid, I owe the author an apology.

Woody Dudley knows enough number theory not to attack the Riemann Hypothesis, and not enough chemical engineering to attack any large open problems in that discipline.