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Gamma. Exploring Euler’s constant. (English) Zbl 1023.11001

Princeton, NJ: Princeton University Press. xxiv, 266 p. (2003).
If a teacher of mathematics wants to offer a present to his students or to him (or her-)self, this book will be the right choice. The story of the gamma constant is a fascinating one, and one can find here plenty of inspiring ideas and connections between various fields of mathematics. The beauty of mathematics is reflected by the many important occurrences in the harmonic world of the harmonic series, Bernoulli numbers, gamma function, continued fraction behaviour of Benford’s law, distribution of primes, the Riemann zeta-function, Pell’s equation, irrationality questions, and many more.
This book is not for professional mathematicians but rather aims at students of mathematics and those who teach them. It is centered on the personality of Euler and the ideas he left for his successors to use and ponder. These ideas are simple enough to be accessible and deep enough to give a feeling for the enchanting beauty of mathematics.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Y60 Evaluation of number-theoretic constants
00A05 Mathematics in general
97-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics education
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Online Encyclopedia of Integer Sequences:

Decimal expansion of Euler’s constant (or the Euler-Mascheroni constant), gamma.
Numbers k such that if 2 <= j < k then the fractional part of the k-th partial sum of the harmonic series is < the fractional part of the j-th partial sum of the harmonic series.
Numbers in base 9.
Numbers n such that the sum of divisors of n^3 is a square.
Numbers n such that sum of divisors of n^2 is a cube.
Numbers n such that n divides Sum_{k = 1..n} A000005(k).
Decimal expansion of Kempner series Sum_{k>=1, k has no digit 1 in base 10} 1/k.
Decimal expansion of Sum_{k >= 1, k has no digit 2 in base 10} 1/k.
Decimal expansion of Sum_{k >= 1, k has no digit 3 in base 10} 1/k.
Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 4 in base 10} 1/k.
Decimal expansion of Kempner series Sum_{k>=1, k has no digit 5 in base 10} 1/k.
Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 6 in base 10} 1/k.
Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 7 in base 10} 1/k.
Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 8 in base 10} 1/k.
Decimal expansion of Kempner series Sum_{k>=1, k has no digit 9 in base 10} 1/k.
Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 0 in base 10} 1/k.
Least k such that H(k) > 10^n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.
Least k such that H(k) > 2^n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.
Least m such that the average number of divisors of all integers from 1 to m equals n, or 0 if no such number exists.
a(n) = least k such that the average number of divisors of {1..k} is >= n.
Decimal expansion of Porter’s Constant.
Numerator of -3*n + 2*(1+n)*HarmonicNumber(n).
Continued fraction for e^gamma.
Denominator of -3*n + 2*(1+n)*HarmonicNumber(n).
Numerators of rationals used in the Euler-Maclaurin type derivation of Stirling’s formula for N!.
Numerator of q_n = -4*n + 2*(1+n)*HarmonicNumber(n).
Decimal expansion of 2*Pi/log(2).
Continued fraction expansion of 2*Pi/log(2).
Decimal expansion of 1/824633702441.