Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
by John Derbyshire
Joseph Henry Press, Washington, D.C.
Copyright 2003, Second Printing June 2003
Hardcover, 422 pages + 18 pages of meta-material
List Price: $27.95
Prologue
I just finished reading this pleasant book. The subject is about Riemann Hypothesis (RH). In the year 1900, the renowned mathematician, David Hilbert gave a famous talk in which he reminded of 23 unsolved problems to be tackled in the newly dawned 20th century. Of these, 22 have, thus far, been solved. Only one has eluded mathematicians and the scientific community. RH is a 150 year old hypothesis that has not been proved or disproved so far. I will describe the problem later in this page. This book excited me to do my own research and come up with the following monograph on RH.
History
The Riemann Hypothesis is currently the most famous unsolved problem in mathematics. Like the Goldbach Conjecture (all positive even integers greater than two can be expressed as the sum of two primes), it seems true, but is very hard to prove. RH currently has several prestigious awards and large cash prizes dedicated to its solution.
Review
John Derbyshire has done a great job in writing this book. He has brought to amateurs a concept that used to be in the realm of professional mathematicians. By bringing the RH to Amateur mathematicians and math lovers, Derbyshire and writers like him who popularize out-of-reach scientific matters (Isaac Asimov, Martin Gardner, etc.) commendably help advance science. It is not all that unlikely for RH to be solved by an amateur mathematician! Remember that it wasn’t the professional computer engineers who built and advanced personal computers, but a few amateur computer enthusiasts who accomplished it in their garages! Or, that one of the greatest mathematicians of all times and the first number theorist was Pierre de Fermat who was an amateur mathematician, while professionally, he was a lawyer and jurist.
It is evident that the Western civilization began its ascent upon Gutenberg's invention of the printing machine which popularized books which were previously accessible by a few, and made them available to all. Writing such books for non-professionals on such deep and specialized subjects as RH is a double bound challenge. But, it makes the subject available to those who seek it. The format of the book is also very interesting. The odd numbered chapters are mathematical and the even numbered ones are biographical and historical. Thus, Derbyshire builds the backdrop against which RH was developed, with discussions of the personas and era, while simultaneously prepares the reader for the trip, then walks him step by step toward the mathematical summit.
Problem Description
The attraction of the Riemann zeta-function lies in its connection to the primes. This connection comes through the Euler product formula, or as Derbyshire refers to it, "The Golden Key." This states that when the real part of s is greater than 1, a magnificent relationship between the natural numbers and primes hold true. In his 1859 paper On the Number of Primes Less Than a Given Magnitude, Bernhard Riemann (1826-1866) examined the properties of the latter mentioned relationship. Zeta function is defines as:
Euler showed that z(2) = p2/6 , and solved all the even integers up to z(26). See the Riemann Zeta Function in the CRC Concise Encyclopedia of Mathematics for more information on this. It is possible for the exponent s to be Complex Number (a + i b). A root of a function is a value x such that f(x) = 0.
This Zeta function is analytic for real part of
Euler had proven this relationship for s being a
real number; Riemann extended it to
This function is analytic at all
points of the complex plane except the point
The Riemann Hypothesis
All non-trivial roots of the Zeta function are of the form (1/2 + i b).
The Functional Equation of the Zeta Function
The functional equation of the zeta function is
Explanation: The zeta
function has no zeros in the region where the real part of
Graphical Representation of the Riemann Hypothesis
Mathematica can plot the Zeta function for complex values, so here are the absolute value of z(1/3 + i b) and z(1/2 + i b).
|z(1/3 + b I)| for b = 0 to 85. This is a cross section of the 3-D Riemann landscape.
Note how the function never dips to zero.
It seems like |z(a + i b)| is bounded away from zero when a doesn't equal 1/2.
|z(1/2 + i b)| for b = 0 to 85. This is a cross section of the 3-D Riemann landscape.
Note how often
the function dips to zero.
(RH: All non-trivial zeros of the zeta function lie on the line
The first 5 zeroes of |z(1/2 + i b)| are at b = 14.1344725, 21.022040, 25.010858, 30.424876, 32.935062
Until May 4 2002, the first ten billion zeroes have been computed and have been shown to be on the critical line, 1/2.
Up to that range, no counter example has been shown that there is a zero outside of the critical line, 1/2. Below, you see the imaginary parts of all the non-trivial zeros where b is less than 100. These comprise the first 29 of the above mentioned 10 billion zeros.
14.134725142
21.022039639
25.010857580
30.424876126
32.935061588
37.586178159
40.918719012
43.327073281
48.005150881
49.773832478
52.970321478
56.446247697
59.347044003
60.831778525
65.112544048
67.079810529
69.546401711
72.067157674
75.704690699
77.144840069
79.337375020
82.910380854
84.735492981
87.425274613
88.809111208
92.491899271
94.651344041
95.870634228
98.831194218
So RH is at least true up to a very high range, but is it always true, in that is there some very high area on the imaginary axis that RH won't be true? Mathematicians are not convinced until an analytical proof be given to lay this to rest!
Below is a 3-D representation of the above graphs. To show zeros, it is easier to show them as truncated infinite values when the zeta function is inversed. The right 3-D graph below shows the zeros as tip of spikes on the critical line, x=1/2. The above 2-D graphs are cross sections of the left graph below.
At the summit, one sees the vast beauty of Riemann's landscape. Many professional and amateur mathematicians find the subject of Riemann Hypothesis so beautiful and so fascinating.
Epilogue:
In the graduate school of electrical engineering (EE), my minor outside of EE was mathematics. I had to take several math courses including Mathematical Analysis as well as Complex Variables. In electrical engineering, many types of mathematical transformations are used to change the problems of electrical signals in time or spatial domains into simpler problems in frequency or spatial frequency domains. Techniques like Fourier Transform, Laplace Transform, Haar Transform, etc. are studied in this field. I thought if the Riemann Hypothesis be transformed into another domain, it could render an easier problem to solve. Thus, I laboriously managed to apply Fourier Transform to RH. What I obtained was certainly a different problem, but just as difficult. Some day, I will apply to RH the other transforms that I have learned. You never know ....!
Bernhard Riemann (1826-1866)
