Mathematics problem that remains elusive — and beautiful

PRIME OBSESSION: BERNHARD RIEMANN AND THE GREATEST UNSOLVED PROBLEM IN

MATHEMATICS

By John Derbyshire

Joseph Henry Press, $27.95, 422 pages, illus.

REVIEWED BY RAYMOND PETERSEN

Bernhard Riemann (1826-1866) published nine scientific articles during his short life. Two were major intellectual blockbusters. The first, presented in 1854 at Gottingen University, qualified him for a tenured position on that school’s soon-to-become world-class mathematics department.

Riemann’s lecture, “On the Hypotheses that Lie at the Foundation of Geometry,” is regarded as one of the 10 top presentations in mathematics, ever. In it, he outlined a new type of geometry, which led to such diverse discoveries as atomic bombs and black holes and today is famously known as Riemann Geometry.

His second blockbuster article is the subject of John Derbyshire’s new book, “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.” That problem, entitled “On the Number of Prime Numbers Less Than a Given Quantity,” was likewise delivered in lecture form, this time in August of 1859 on the occasion of Riemann’s induction into the Berlin Academy of Science, a tremendous honor of the then 32-year-old mathematician.

The title of the second great lecture is deceptively simple. In that paper, Riemann proposed a mechanism, a modified zeta function which looks like this, _(s) = , for determining the distribution of prime numbers. He also formulated a conjecture, the Riemann Hypothesis — all non-trivial zeros (of the Riemann Zeta Function) have real parts one half. A proof of this hypothesis would validate the use of the Riemann Zeta Function for determining prime number distribution.

Mr. Derbyshire calls the ongoing proof (or disproof ) of the Riemann Hypothesis the “greatest unsolved problem in mathematics” — and with justification: Mathematicians have fiercely debated his proposal ever since its introduction a century and a half ago. But to the innumerate, the significance of the debate — and its intensity — must be incomprehensible. Most people have learned at some point in their math education what prime numbers are: They’re those irreducible number — such as 5, 13, 23 — that are factors only of themselves and the number one.

Why should a hypothesis which purports to be able to determine the distribution of prime numbers among all possible numbers have such earth-shaking influence? For mathematicians the pure intellectual enjoyment yielded by contemplation of the Riemann Hypothesis helps explain its great attraction.

But the significance of Riemann’s work goes beyond that. For centuries, mathematicians have argued that math offers us ways to comprehend reality far more accurately than any other tool, certainly more accurately than mere language, which mathematicians have almost always tended to find messy and inexact.

“Numbers are the thoughts of God,” is a saying attributed to Pythagoras, the great 6th century B.C. Greek mathematician and author of his own theorem — the Pythagorean — that’s played a major part in the evolution of mathematics, and is itself a description of reality based on the relationship among numbers.

For Mr. Derbyshire, the Riemann Hypothesis supports the long-time belief in mathematics as the key to understanding reality. He cites the example of an afternoon tea in 1972 held at the Institute for Advance Study, the site of many 20th century mathematical and scientific breakthroughs.

On the occasion described by Mr. Derbyshire, Hugh Montgomery, a graduate student was chatting with the great physicist Freeman Dyson, when Montgomery happened to mention his findings on the distribution of prime numbers based on the Riemann’s zeta function.

Dyson, one of the most highly-regarded scientists of his time, poignantly informed the young man that his findings into the distribution of prime numbers corresponded with the spacing and distribution of energy levels of a higher-ordered quantum state.

Wow! Thirty years ago, during tea and conversation, the numbers flowing out of Riemann’s zeta function, founded on an unproven hypothesis, find themselves intimately associated with a fundamental property of matter. Yet once again, reality can be seen as logically and mathematically designed, and comprehensible with numbers, properly used.

Mr. Derbyshire explains that it is because so much of modern mathematics and an important aspect of quantum physics depend on the Riemann Hypothesis being true that so many minds and careers have been devoted to investigation Riemann’s zeta function for determining the distribution of prime numbers.

What those minds hope to determine is proof (or, as the author remarks, heaven forbid, disproof) of the Riemann Hypothesis which supports the use his zeta function for prime number distribution. It’s not the author’s aim to solve the puzzle, but to trace the Riemann Hypothesis’s significance and show how it became central to mathematics.

For those interested in the mathematics of the Riemann Hypothesis, a background in algebra, geometry, and introductory calculus should be sufficient. But surely most everyone can enjoy Mr. Derbyshire’s lucid and informatively anecdotal description of the thinkers who contributed to our understanding of prime numbers.

They are all extraordinary people. Leonhard Euler (1707-1783) was only 20 when he left his native Switzerland to go to St. Petersburg, to become one of those Western Europeans brought to Russia by Peter the Great to bring his country abreast of the scientific developments of the West.

Euler made major contributions to the mathematics of prime numbers. So did the amazing Carl Friedrich Gauss (1777-1855), the brilliant son of landless German peasants and the young Riemann’s major professor at Gottingen. Mr. Derbyshire tells the story that Gauss, when he was only 12, responded within a minute to a teachers query, “What is the sum of all whole numbers from 1 to 100?”

Gauss answered, correctly, that the sum was 5,050. How had he arrived at the correct answer?, the surprised teacher asked. Gauss answered that he had quickly reasoned that if all 100 numbers are listed in two superimposed rows, one going 1 to 100 and the other 100 to 1, placed below, and then each column is added, the sum is always 101.

Since there are 100 such additions, the total sum of the 100 columns is 10,100. But the number 10,100 is twice the value that is needed, because of the duplication of sums. Therefore, by dividing 10,100 by 2, the correct answer, 5050, is obtained.

This from the 12-year-old Gauss. The mathematicians who followed Riemann described by Mr. Derbyshire were similarly brilliant — and eccentric. David Hilbert (1862-1943), for example, succeeded Riemann, in the great mathematician’s chair at the University of Gottingen.

The perfect example of the absent-minded professor, Hilbert on one occasion was asked to given the funeral oration of a student of his who died an untimely death. Hilbert proceeded with the oration, but a short way into his talk mentioned the student had worked on the Riemann Hypothesis — and then proceeded to give the uncomprehending crowd of mourners a full-blown lecture on Riemann’s work on prime numbers and on the complexity and beauty of the Riemann Hypothesis.

On another legendary occasion, Hilbert inquired of his class in advanced mathematics why a certain student had not been in class for some time. He was told that the student in question had left the university to become a poet. “I can’t say I’m surprised,” the mathematics professor is said to have observed, “I never thought he had enough imagination to be a mathematician.”

But Hilbert was a brilliant man who delivered in 1900 a highly-regarded address at the Second International Congress of Mathematics. In the talk he summarized the direction mathematics should take in the 20 th century and underlined the importance of the Riemann Hypothesis.

It is interesting that in the early part of that century, two insurance company executives with an avocation in mathematics added to the world’s understanding of Riemann’s great work. Jorgen Gramm of Denmark and Harold Cramer of Sweden were rough contemporaries of two similarly creative American insurance company executives with avocations in other fields, Charles Ives, the composer, and the poet Wallace Stevens.

It is Stevens who seems to have the closest affinity with the kind of love of harmony and order sought by the great mathematicians, despite Hilbert’s disparagement of the poetic imagination. It is easy to see a mathematician’s passionate need to find proof for the Riemann hypothesis summarized metaphorically in the penultimate stanza of Stevens’ great poem, “the Idea of Order at Key West.”

Ramon Fernandez, tell me, if you know,

Why, when the singing ended and we turned

Towards the town, tell why the glassy lights,

The lights in the fishing boat at anchor there,

As the night descended, titling in the air,

Mastered the night and portioned out the sea,

Fixing emblazoned zones and fiery poles,

Arranging, deepening, enchanting night.

John Derbyshire’s “Prime Obsession” is an intellectual tour de force and an excellent read.

Raymond Petersen is a botanist and Professor of Biology at Howard University. He teaches a course in the History and Philosophy of Science. rpetersen@howard.edu.