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Gamma: Exploring Euler's Constant

Julian Havil
Foreword by Freeman Dyson
Copyright Date: 2003
Pages: 304
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  • Book Info
    Book Description:

    Among the myriad of constants that appear in mathematics,p,e, andiare the most familiar. Following closely behind isg, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery.

    In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics.

    Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . up to 1/n, minus the natural logarithm ofn--the numerical value being 0.5772156. . .. But unlike its more celebrated colleaguespande, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction.

    Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!).

    Sure to be popular with not only students and instructors but all math aficionados,Gammatakes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.

    eISBN: 978-1-4008-3253-8
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xiv)
  3. Foreword
    (pp. xv-xvi)
    Freeman Dyson

    I am delighted to be allowed to add a few words to this book by Julian Havil, who is a teacher of mathematics at the school where I was a student sixty years ago. I fell in love with mathematics at the school and have been a professional mathematician ever since.

    This book is not for professional mathematicians but rather it is aimed at students of mathematics, be they eager high school students or undergraduates, and those who teach them. It is an inspiring book that will give them an idea of how enchanting mathematics can be.

    Mathematics is often...

  4. Acknowledgements
    (pp. xvii-xviii)
  5. Introduction
    (pp. xix-xxiv)

    It is tempting to think that there are just three special mathematical constants: π,eandi. In fact there are many, each with its own definition, each originating in some natural way in its own area of mathematics, each given a special symbol and a name too. They need symbols to represent them because they are awkward; that is, they have no convenient, finite numeric representation and no patterned infinite one: the ratio of the circumference to the diameter of any circle is not 3.142 or${\textstyle{{22} \over 7}}$, it is 3.141 59. . . , which is as mysterious as...

  6. CHAPTER ONE The Logarithmic Cradle
    (pp. 1-20)

    In an age when a ‘computer’ is taken to mean a machine rather than a person and calculations of fantastic complexity are routine and executed at lightning speed, constricting difficulties with ordinary arithmetic seem (and are) extremely remote. The technological freeing of mathematics from the manacles of calculation is very easy to take for granted, although the freedom has been newly won; as recently as the mid 1970s, a mechanical calculator, slide rule or table of logarithms would have been used to perform anything other than the most basic calculations and the user would have been grateful for them. In...

  7. CHAPTER TWO The Harmonic Series
    (pp. 21-26)

    On 11 July 1382, in the beautiful Norman city of Lisieux, Nicholas Oresme died at the age of 59; he had been the city’s bishop since 1377. Born into the Late Middle Ages (in Allemagne in 1323), his scholarship extended from the development of the French language to taxation theory and his distinguished career included the Deanship of Rouen and being chaplain to King Charles V of France, for whom he translated Aristotle’sEthics, Politics and Economics. He taught the heliocentric theory of Copernicus over 100 years before Copernicus was born and suggested graphing equations nearly 200 years before the...

  8. CHAPTER THREE Sub-Harmonic Series
    (pp. 27-36)

    The incredibly slow divergence of${H_n}$suggests that we would not need to alter its terms by much to force convergence, and by altering we mean omitting or cancelling. In this chapter, we will attempt just that.

    If we start taking out terms in a structured way, we might start with

    $\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \cdots = \frac{1}{2}\left( {1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots } \right)$,

    or alternatively

    $1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \cdots > \;1\; + \;\frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \cdots = 1 + \frac{1}{2}\left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots } \right)$,

    both of which clearly diverge, which will have implications on p. 102.

    So removing ‘half’ of the terms is not enough to force the depleted series to converge, nor would a third or any other fraction of it. Taking only powers of any single number...

  9. CHAPTER FOUR Zeta Functions
    (pp. 37-46)

    It is time to look at one of the ‘advanced’ functions of mathematics and one which lies at the core of the study of analytic number theory; a function which, according to M. C. Gutzwiller, ‘is probably the most challenging and mysterious object of modern mathematics’. We will see it here in its own right and, in Chapter 6, linked to a second ‘advanced’ function and again in the final chapter, where its deepest behaviour is the stuff of the Riemann Hypothesis.

    The series

    $\sum\limits_{r = 1}^\infty {\,\frac{1}{{{r^2}}}} = 1 + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \cdots $

    holds a special place in mathematical lore. A simple calculation suggests that it converges to...

  10. CHAPTER FIVE Gamma’s Birthplace
    (pp. 47-52)

    So, the harmonic series diverges, slowly. Just how slowly can be measured using its interpretation as a discrete logarithm. The area$\smallint _1^n(1\,/\,x)\,{\rm{d}}x = {\rm{ln}}\;n$is bounded below by the areas of the underestimating rectangles and above by the areas of the overestimating rectangles, which using Figures 5.1 and 5.2 results in the inequality

    $\frac{1}{2}\; + \;\frac{1}{3}\; + \; \cdots + \;\frac{1}{n} < \int_1^n {\frac{1}{x}\,{\rm{d}}x\; < \;1\; + \;\frac{1}{2}\; + \;\frac{1}{3} + \cdots + \frac{1}{{n - 1}}} $,


    ${H_n} - 1\; < \;{\rm{ln}}\;n\; < \;{H_n} - \frac{1}{n}$


    ${\rm{ln}}\,n\; + \;\frac{1}{n} < \;{H_n} < \;{\rm{ln}}\;n\; + \;1$.

    We have an estimate of${H_n}$as lnnwith an error of at least$1/n$and at most 1, with${H_n}$confined between the curves, as shown in Figure 5.3. Put another way,

    $\frac{1}{n} < \;{H_n} - {\rm{ln}}\;n\; < \;1$

    and so, if the limit exists,$0 \le \;{\rm{li}}{{\rm{m}}_{n \to \infty }}({H_n} - {\rm{ln}}\,n)\; \le \;1$.

    If we...

  11. CHAPTER SIX The Gamma Function
    (pp. 53-60)

    We will now look at that second ‘advanced’ function and its link with Euler’s constant and with the Zeta function.

    The striking integral

    $\int_0^1 {\,{{\left( {{\rm{ln}}\left( {\frac{1}{t}} \right)} \right)}^{x - 1}}{\rm{d}}t} $

    occupied some of Euler’s many mathematical thoughts during the years 1729 and 1730 and in a letter to Christian Goldbach (1690–1764), dated 8 January 1730, he proposed its use in a quite startling way. It converges for$x\; > \;0$and can be considered as a function of x in that domain, a function whose properties are surprising and unexpectedly useful. In 1809 Adrien-Marie Legendre (1752–1833) gave it the name Gamma and the matching symbol Γ...

  12. CHAPTER SEVEN Euler’s Wonderful Identity
    (pp. 61-64)

    Euler wanted to establish the divergence of the reciprocals of the primes. We have already seen Erdös’s stylish proof of this but that will not prevent us from revelling in the glory of Euler’s inventiveness, particularly as it brought about a result which is the cornerstone of analytic number theory, and which we will have considerable use of later.

    The positive integers are a Unique Factorization Domain, that is, every positive integer is uniquely expressible as a product of primes (which is why 1 is not considered prime), and from this innocent fact Euler extracted wonder by producing the equivalent...

  13. CHAPTER EIGHT A Promise Fulfilled
    (pp. 65-68)

    Earlier we mentioned the barely credible result that the probability of two randomly chosen integers being co-prime is$1\,:\,{\textstyle{1 \over 6}}{\pi ^2}$. With Euler’s formula, combined with several other mathematical tools listed below, we are able to prove the fact; but first those tools.

    (1) In set theory, the symbols$ \cap $and$ \cup $(respectively, the intersection and union of sets) are defined to be the set of all elements common to both and contained in either or both, respectively. These ‘binary’ operations on sets give rise to an algebra, known as ‘Boolean algebra’, named after the English mathematician George Boole (1815–1864), from...

  14. CHAPTER NINE What Is Gamma . . . Exactly?
    (pp. 69-80)

    We have pretty convincing evidence that the constant$\gamma $exists, but no precise proof. Euler did not live in an age of great mathematical rigour and he assuredly was not given to spending his days trying to prove what seemed to him to be intuitively obvious: such thoroughness was to be the stuff of the 19th century. In the 21st, we would be uncomfortable without the security of knowledge that$\gamma $really does exist and so we will deal with that matter now.

    Given that$\gamma $does exist, perhaps the first thing to notice is that we seem to have...

  15. CHAPTER TEN Gamma as a Decimal
    (pp. 81-90)

    Our earlier focus on the Zeta series has meant that, in terms of the summation of series, we have in a way started on the second rung of the ladder, with the first occupied by the family${1^k} + {2^k} + {3^k} + \cdots + {n^k}$for$k \in N$. In 1784, at the age of seven, Gauss had famously summed the integers from 1 to 100 in seconds (to the amazement of his teacher) when he noticed that the series could be thought of as 50 pairs of numbers each summing to 101; of course, the young genius could not have known that the ancient Greeks, Hindus and Arabs...

  16. CHAPTER ELEVEN Gamma as a Fraction
    (pp. 91-100)

    It is a simple matter of arithmetic to use the decimal approximations of a number to generate fractional approximations of it. For example,

    $\gamma = 0.577\;215\;664\;901\;532\;860\;606\;5\, \ldots $

    results in the approximations:

    $\frac{5}{{10}},\;\frac{{57}}{{100}},\;\frac{{577}}{{1000}},\;\frac{{5772}}{{10\,000}},\;\frac{{57\,721}}{{100\,000}},\; \cdots = \frac{1}{2},\;\frac{{57}}{{100}},\;\frac{{577}}{{1000}},\;\frac{{2881}}{{5000}},\;\frac{{57\;721}}{{100\,000}},\, \cdots \cdot $

    Yet, compare the accuracy of the approximations with the mysterious sequence

    $\frac{3}{5},\;\frac{4}{7},\;\frac{{11}}{{19}},\;\frac{{15}}{{26}},\;\frac{{71}}{{123}},\;\frac{{228}}{{395}},\;\frac{{3035}}{{5258}},\; \cdots \cdot $

    And what about${\textstyle{{323\,007} \over {559\,595}}}\,?$These perplexing numbers are progressively more accurate approximations to$\gamma $and better than any comparable fraction arising as above. If we do want to approximate$\gamma $by fractions, we would do well to look to them. The question is, where do they come from?

    Fermat was given to posing number-theoretic problems. The most famous of...

  17. CHAPTER TWELVE Where Is Gamma?
    (pp. 101-118)

    Gamma’s definition$\gamma \; = \;{\text{li}}{{\text{m}}_{n \to \infty }}({H_n} - {\text{ln}}\,n)$, when rewritten as the asymptotic approximation${H_n} \approx \;{\text{ln}}\;n\; + \;\gamma $, provides a simple (and accurate) method for approximating the partial sums of the harmonic series. The lack of an explicit formula for${H_n}$together with its glacially slow divergence makes the approximation all the more important and with that approximation we have an inevitable appearance of$\gamma $; already we have seen the estimate used on a number of occasions. Its connection with the Gamma function guarantees$\gamma $’s role in analysis and the Gamma function’s connection with the Zeta functions guarantees$\gamma $’s role in number theory. The number is inevitably, intrinsically...

  18. CHAPTER THIRTEEN It’s a Harmonic World
    (pp. 119-138)

    We will now take a brief look at several of the ways in which${H_n}$appears, and the pattern of numbers$1,\;\tfrac{1}{2},\;\tfrac{1}{3},\; \ldots $forming its terms appear, in some areas of considerable diversity. The selection is by no means comprehensive and each initiative can be developed (in some cases very considerably) beyond where we leave it, but to delve deeper or to embrace more widelywould engulf more pages than this book could afford. Firstly, though, we ought to address the question of the name ‘harmonic’.

    With two numbersaandb, if one had to write down three examples of an...

  19. CHAPTER FOURTEEN It’s a Logarithmic World
    (pp. 139-162)

    As we have mentioned in the Introduction, the reader of this book will need little convincing that logarithms appear with great frequency in mathematics and its applications, particularly with so many differential equations involving them or exponentials in their solution. Power laws abound in nature: Kepler’s third law, the universal law of gravitation, Boyle’s Law, etc.Abrowse through any science book will yield any number of examples, and where there is a power law, there is a linearizing logarithm, as Kepler may have experienced. The intensity of earthquakes is measured on the logarithmic Richter scale, fractal dimension is defined in terms...

  20. CHAPTER FIFTEEN Problems with Primes
    (pp. 163-188)

    Prime numbers have appeared several times in this book. Their study has long held centre stage in number theory and their behaviour, at times seemingly so undisciplined, can sometimes appear determined by an unknown, powerful authority unwilling to disclose its design. The leading quotation makes evident the great Euler’s frustration; Erdös, paraphrasing Einstein, said ‘God may not play dice with the Universe, but there’s something strange going on with the prime numbers!’ and R. C. Vaughan spoke for many when he said, ‘It is evident that the primes are randomly distributed but, unfortunately, we don’t know what random means.’ Three...

  21. CHAPTER SIXTEEN The Riemann Initiative
    (pp. 189-216)

    In his paper Riemann considered another weighted prime counting function, which we will write as$\Pi (x)$, related to the harmonic series and defined by

    $\Pi (x)\; = \;\sum\limits_{\scriptstyle {p^r} < x, \atop \scriptstyle p\;{\rm{prime}}} {\frac{1}{r}} $,

    which again reveals a bit more about itself if we look at a couple of examples:

    $\Pi (20) = \sum\limits_{\scriptstyle {p^r} < 20, \atop \scriptstyle p\;{\rm{prime}}} {\frac{1}{r}} $

    $ = \left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right) + \;\left( {\frac{1}{1}\; + \;\frac{1}{2}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)$,

    where the bracketing is by the primes 2, 3, 5, . . . , 19, and

    $\Pi (30) = \sum\limits_{\scriptstyle {p^r} < \,30, \atop \scriptstyle p\;{\rm{prime}}} {\frac{1}{r}} $

    $ = \left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right)\; + \;\left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3}} \right)\; + \;\left( {\frac{1}{1}\; + \;\frac{1}{2}} \right) + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)$,

    where the bracketing is by the primes 2, 3, 5, . . . , 29.

    These can be rewritten as

    $\Pi (20) = \left( {\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}} \right) + \;\frac{1}{2}\left( {\frac{1}{1}\; + \;\frac{1}{1}} \right)\; + \;\frac{1}{3}\,\left( {\frac{1}{1}} \right)\; + \;\frac{1}{4}\;\left( {\frac{1}{1}} \right)$


    $\Pi (30) = \left( {\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}} \right) + \;\frac{1}{2}\left( {\frac{1}{1} + \frac{1}{1} + \frac{1}{1}} \right) + \frac{1}{3}\left( {\frac{1}{1} + \frac{1}{1}} \right) + \frac{1}{4}\left( {\frac{1}{1}} \right)$.

    The first bracket just counts the primes less than the number, the second those less than...

  22. APPENDIX A The Greek Alphabet
    (pp. 217-218)
  23. APPENDIX B Big Oh Notation
    (pp. 219-220)
  24. APPENDIX C Taylor Expansions
    (pp. 221-224)
  25. APPENDIX D Complex Function Theory
    (pp. 225-248)
  26. APPENDIX E Application to the Zeta Function
    (pp. 249-254)
  27. References
    (pp. 255-258)
  28. Name Index
    (pp. 259-262)
  29. Subject Index
    (pp. 263-266)