» Gamma: Exploring Euler's Constant (Princeton Science Library)

Gamma: Exploring Euler's Constant (Princeton Science Library) Reviews

Customer Rating:
Summary: One of the best popular math books
Comment: Gamma: Exploring Euler's Constant (Princeton Science Library)
I find many popular math books not very satisfying because they either don't provide advanced enough math subject matter, or their proofs are too sketchy or not intuitive.
Havil's book offers a good balance of entertaining biographical and historical facts, and mathematical material advanced just enough to provide "meat" without it being out of reach for most readers.

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Customer Rating:
Summary: The transcendence problem
Comment: I have always found Euler's constant interesting because I would like to be able to say that it is one of the 3 great transcendental numbers along with pi and e. The number e is what Kurt Mahler called an S number; perhaps pi and Euler's constant also are. My interest started about 1968 and I was soon led to the gamma and Riemann zeta functions. I am pleased to see that Havil confirms the path I followed.
This is a very interesting book about a relatively unpublicized development of calculus. Admittedly this book requires good skills in mathematics, but let's be candid: a good understanding of mathematics calls for experience in working things out.
Now I must express a disappointment. I bought the book at least in part because I wanted to know the history of attempts at proving (or disproving) the transcendence of Euler's constant. I have found almost nothing. There is a copious history of attempts at proving Fermat's last theorem. Perhaps Euler's constant has just not attracted so very much effort. I do not know of any financial reward that has been offered for Euler's constant. Hilbert in 1900 did not specify this number in his seventh problem. I have not seen this problem emphasized as one for the 21st century.
For some years I thought of questions that might lead to a proof or disproof of transcendence. How long is the shortest proof that Euler's constant is transcendental or not? Can the question be answered in finitely many steps? Is there a number related to Euler's constant in the way that e is related to pi? I did not find these questions mentioned in this book.
It is my hope that some high school student (or even grade school student) will read this book and, perhaps after some 40 years, become the Andrew Wiles of Euler's constant.

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Customer Rating:
Summary: An excellent elementary overview of analytic number theory.
Comment: This book introduces gamma and zeta functions, an elementary overview of analytic number theory, and a few applications of harmonic series.

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Customer Rating:
Summary: Gamma Exploring Euler's Constant
Comment: I find this book very interesting, informative and pleasant.
The content is ample. It touches many matters concerning Logarithms, Harmonic Series, Gamma constant ,Primes, Z function and Riemann Hypothesis.
However, the lecture it's not easy because it was written at a college level.
Notwithstanding I recommend the book for those who love Number Theory.
I hope that in a future edition I will see corrected, the bizarre error in page 195, line 4 (from the bottom). It says: "...the imaginary part of each of the complex numbers is always 0.5"
It must say "... the real part..."
This error is repeated in page 196 Table 16.1. In that table all the + 0.5i must be changed to + 0.5 (without i).
Also, the third chapter: Sub-Harmonic Series will be more complete if to it, the author adds the important theorem:

Lim. Sum k=2^n to k=2^(n+1){1/k} = Log(2)

Ludovicus

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Customer Rating:
Summary: Good but demanding
Comment: Target:

Despite what the author says in the introduction, the book is addressing people with a firm grip on high-school (=real) calculus; not only does Havil go into difficult topics, his proofs are rather succinct and often require some thinking on the reader's part. However, a good high-school student should be able to follow most of the book, even the last chapter that deals with complex analysis as Havil does a great job explaining it.


Pluses:

I think this book is ideal for high-school students and undergrads who want to know more mathematics in general and gamma and real analysis in particular.
It's one of the best popular books i've read. Havil presents difficult issues with great ease, leaving tiny bits of proofs for the reader to fill in, but which shouldn't be a problem for anyone who was able to understand what he did up to that point (as i have said, high-school calculus should be enough).
Something i appreciated is Havil states and proves a LOT of exciting results like the probability that two numbers should be co-prime is 6/pi*2, Euler's product formula, etc. The writing is good, clear and direct, Havil delivers on every promise he makes and doesn't do a lot of hand-waving like most other popular math books do; however, in chapter 12, he writes 12 formulas that link gamma with pi, e, log(2), pi*2, the floor function, etc. and leaves 10 of them for the reader to prove.
The are cases when the author deliberately chose a longer proof to illustrate how incredibly close some mathematical expressions are (for ex. he shows that 1+1/2+...+1/n -log(n+1) is bounded by zeta(2)=pi*2/6).
Havil makes use of a lot of historical information on the mathematical concepts involved, as well as the people who developed them, and he does it in the same thorough manner in which he wrote the book. Great info here as well.


Minuses:

I have found about 20 typos throughout the book, including 4 in formulas used in proofs, although nothing that can not be corrected by simply checking the next line for continuity.
The book is not all about gamma; as Havil says, gamma is deeply connected to the harmonic series and to the logarithms so a closer look at these and their other functions is necessary; however, in some cases, i felt the author had strayed a bit too much.

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