EULER–MASCHERONI CONSTANT

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The 'Euler–Mascheroni constant' is a mathematical constant, used mainly in number theory. It is defined as the limiting difference between the harmonic series and the natural logarithm:
:$gamma\; =\; lim\_\{n$
ightarrow infty } left[ left(
sum_{k=1}^n rac{1}{k}
ight) - log(n)
ight]=int_1^inftyleft({1overlfloor x
floor}-{1over x}
ight),dx
It is usually denoted by the lowercase Greek letter γ (gamma). Sometimes it is called simply the ''Euler constant'', though it ought not to be confused with ''e'', which is often called ''Euler's number''.
Its numerical value to 50 decimal places is
0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 ...
{| border="1" style="float: right; border-collapse: collapse;"
| colspan="2" align="center" | List of numbers
γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ
|-
|Binary
| 0.1001001111000100011...
|-
| Decimal
| 0.5772156649015328606...
|-
| Hexadecimal
| 0.93C467E37DB0C7A4D1B...
|-
| Continued fraction
| $0\; +\; cfrac\{1\}\{1\; +\; cfrac\{1\}\{1\; +\; cfrac\{1\}\{2\; +\; frac\{1\}\{1\; +\; frac\{1\}\{\; ddots\; \{\}\}\}\}\}\}$
Note that this continued fraction is not periodic.
|}

Contents

History

Properties

Relations to special functions

Asymptotic expansions

''e'' to the power of γ

Generalizations

Cultural appearances

Appearances

Known digits

References

History

The constant was first defined by Swiss mathematician Leonhard Euler in a paper ''De Progressionibus harmonicis observationes'' published in 1735 (Eneström Index 43). Euler used the notation ''C'' and ''O'' for the constant, and initially calculated its value to 6 decimal places. In 1781 he extended this calculation, publishing a value to 16 decimal places.
In 1790 Italian mathematician Lorenzo Mascheroni introduced the notation ''A'' for the constant, and attempted to extend Euler's calculation still further, to 32 decimal places, although subsequent calculations showed that he had made errors in the 20th-22nd decimal places. (From the 20th digit, Mascheroni calculated 1811209008239, while the correct value is 0651209008240.) Mascheroni never used the notation γ. It is from a later time in connection with the Gamma function.^[1]
It is not known whether γ is a rational number or not. However, continued fraction analysis^[2] shows that if γ is rational, its denominator must be greater than 10²⁴²⁰⁸⁰.
In December 2006, Alexander J. Yee, an undergraduate at Northwestern University, set a record for calculating the Euler-Mascheroni constant to 116,580,041 decimal places.^[3]

Properties

As first discovered by Euler, the limit definition can be restated as an explicit series
:
ight)
ight].
The constant is given by several integrals:
:$gamma\; =\; -\; int\_0^infty\; \{\; e^\{-x\}\; log\; x\; \},dx$
::
ight ) },dx
::
ight )e^{-x} },dx
::
ight ) },dx.
Other integrals that include $gamma$ are:
:$int\_0^infty\; \{\; e^\{-x^2\}\; log\; x\; \},dx\; =\; -\; frac14(gamma+2\; log\; 2)\; sqrt\{pi\}$
:
One can express $gamma$ also as a double integral (Sondow 2003, 2005) with equivalent series:
:
ight ).

An interesting comparison by J. Sondow (2005) is the double integral and alternating series
:
ight ) = int_{0}^{1}int_{0}^{1} rac{x-1}{(1+x,y)log(x,y)} , dx,dy = sum_{n=1}^infty (-1)^{n-1} left( rac{1}{n}-lograc{n+1}{n}
ight).
It shows that
ight ) may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series (see Sondow 2005 #2)
:
:
ight )
where $N\_1(n)$ and $N\_0(n)$ are the number of 1's and 0's, respectively, in the base 2 expansion of $n$.
The series for $gamma$ is equivalent to Vacca's interesting 1910 sum
:$$
gamma = sum_{k=2}^infty (-1)^k rac{ left lfloor log_2 k
ight
floor}{k}
= frac12- frac13
+ 2left( frac14 - frac15 + frac16 - frac17
ight)
+ 3left( frac18 - dots - frac1{15}
ight) + dots

where $log\_2$ is the logarithm to the base 2 and $left\; lfloor\; ,$
ight
floor is the floor function.

But Vacca was not the first.
In 1897 Nielsen found this in the form

floor}{k+1} .

In 1926 Vacca found a similar one:
$$
gamma + zeta(2) = sum_{k=1}^{infty} rac1{klfloorsqrt{k}
floor^2}
= 1 + frac12 + frac13 + frac14left( frac14 + dots + frac18
ight)
+ frac19left( frac19 + dots + frac1{15}
ight) + dots


or

$$
gamma = sum_{k=2}^{infty} rac{k - lfloorsqrt{k}
floor^2}{k^2lfloorsqrt{k}
floor^2}
= frac1{2^2} + frac2{3^2}
+ frac1{2^2}left( frac1{5^2} + frac2{6^2} + frac3{7^2} + frac4{8^2}
ight)
+ frac1{3^2}left( frac1{10^2} + dots + frac6{15^2}
ight) + dots


(see Krämer, 2005)
Vacca's series may be obtained by manipulation of Catalan's 1875 integral (see Sondow and Zudilin)
:
The continued fraction expansion of $gamma$ is:
:$gamma\; =\; [0;\; 1,\; 1,\; 2,\; 1,\; 2,\; 1,\; 4,\; 3,\; 13,\; 5,\; 1,\; 1,\; 8,\; 1,\; 2,\; 4,\; 1,\; 1,\; 40,\; ...],$ .

Relations to special functions

$gamma$ can also be expressed as an infinite sum with terms involving the values of the Riemann zeta function at positive integers:
:
:
ight ) + sum_{m=1}^{infty} rac{(-1)^{m-1} zeta(m+1)}{2^m (m+1)}.
Other Zeta-related series include
:
::
ight )
ight ].
::
ight )
ight ]
The error term in last identity is a rapidly decreasing function of ''n''. As a result, the formula is well-suited to efficiently computing the constant to high precision.
A limit related to the Beta function (in terms of Gamma functions) is
:
ight ].
Other interesting limits equaling the Euler-Mascheroni constant are the antisymmetric limit (Sondow, 1998)
:
ight ) = lim_{s o 1} left ( zeta(s) - rac{1}{s-1}
ight )
and
:
ight )
ight ]
::
ight
ceil - rac{n}{k}
ight ).
Closely related to this is the rational zeta series expression. By peeling off the first few terms of the series above, one obtains an estimate for the classical series limit:
:
sum_{m=2}^infty rac{zeta (m,n+1)}{m}
where $zeta(s,k)$ is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, H_n. Expanding some of the terms in the Hurwitz zeta function gives:
:$$
H_n = log n + gamma + rac {1} {2n} - rac {1} {12n^2} + rac {1} {120n^4} - arepsilon , where
The constant can also be calculated as a derivative of Euler's Gamma function:
:$gamma\; =\; -Gamma\text{'}(1).$

Asymptotic expansions

Asymptotic formulas for Euler's gamma are given by (Where $H\_n$ is the ''n''th harmonic number.)
:$gamma\; sim\; H\_n\; -\; ln\; left(\; n$
ight) - rac{1} + rac{1} - rac{1} + ...
:(''Euler'')
:
ight)
:(''Negoi'')
:
ight) + ln left( {n + 1}
ight)}}{2} - rac{1}{{6nleft( {n + 1}
ight)}} + rac{1}{{30n^2 left( {n + 1}
ight)^2 }} - ...
:(''Cesaro'')
The third formula is also called the Ramanujan expansion.

''e'' to the power of γ

The constant ''e''^γ is also important in number theory. Occasionally, ''e''^γ is denoted $gamma\text{'}$ It is expressed with the following limit, where ''p''_''n'' is the ''n''-th prime number:
:$$
e^gamma = lim_{n o infty} rac {1} {log p_n} prod_{i=1}^n rac {p_i} {p_i - 1} ,

which is a restatement of the third of Mertens' theorems. The numerical value of ''e''^γ is:
:$e^gamma\; =1.78107241799019798523650410310717954916964521430343dots$
Other infinite products relating to ''e''^γ include
:
ight )^n
:
ight )^n.
Both of these products result from the Barnes G-function.
We also have
:
ight )^{1/2} left (rac{2^2}{1 cdot 3}
ight )^{1/3} left (rac{2^3 cdot 4}{1 cdot 3^3}
ight )^{1/4}
left (rac{2^4 cdot 4^4}{1 cdot 3^6 cdot 5}
ight )^{1/5} cdots
where the ''n''th factor is the (''n''+1)st root of
:$prod\_\{k=0\}^n\; (k+1)^\{(-1)^\{k+1\}\{n\; choose\; k\}\}.$
This infinite product is due to J. Ser (1926). It was rediscovered by J. Sondow (2003) using hypergeometric functions.

Generalizations

''Euler's generalized constants'' are given by
:
ight],
for 0 < α < 1, with γ as the special case α = 1.^[4] This can be further generalized to
:$c\_f\; =\; lim\_\{n\; o\; infty\}\; left[\; sum\_\{k=1\}^n\; f(k)\; -\; int\_1^n\; f(x)\; ,\; dx$
ight]
for some arbitrary decreasing function ''f''. For example,
:
gives rise to the Stieltjes constants, and
:$f\_a(x)\; =\; x^\{-a\}$
gives
:
where again the limit
:
ight]
appears.
A two-dimensional limit generalization is the Masser–Gramain constant.

Cultural appearances

★ The Euler-Mascheroni constant was used in the solution to the Car Talk puzzler for the week of 23 October 2006. The question involves a car which repeatedly slows down as it nears its destination. The answer presented on the popular radio show uses the Euler-Mascheroni constant.

Appearances

The Euler-Mascheroni constant appears, among other places, in:

★ an inequality for Euler's totient function

★ the growth rate of the divisor function

★ a product formula for the gamma function

★ calculations of the digamma function

★ calculation of the Meissel-Mertens constant

★ expressions involving the exponential integral

★ the first term of the Taylor series expansion for the Riemann zeta function, where it is the first of the Stieltjes constants

★ the third of Mertens' theorems.

★ the Laplace transform of the natural logarithm

Known digits

The number of known digits of the Euler-Mascheroni constant γ has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.^[5]
{| class="wikitable" style="margin: 1em auto 1em auto"
|+ 'Number of known decimal digits of ''γ'' '
! Date || Decimal digits || Computation performed by
|-
| 1734 || 5 || Leonhard Euler
|-
| 1736 || 15 || Leonhard Euler
|-
| 1790 || 19 || Lorenzo Mascheroni
|-
| 1809 || 24 || Johann G. von Soldner
|-
| 1812 || 40 || F.B.G. Nicolai
|-
| 1861 || 41 || Oettinger
|-
| 1869 || 59 || William Shanks
|-
| 1871 || 110 || William Shanks
|-
| 1878 || 263 || John C. Adams
|-
| 1962 || 1,271 || Donald E. Knuth
|-
| 1962 || 3,566 || D.W. Sweeney
|-
| 1977 || 20,700 || Richard P. Brent
|-
| 1980 || 30,100 || Richard P. Brent & Edwin M. McMillan
|-
| 1993 || 172,000 || Jonathan Borwein
|-
| 1997 || 1,000,000 || Thomas Papanikolaou
|-
| December 1998 || 7,286,255 || Xavier Gourdon
|-
| October 1999 || 108,000,000 || Xavier Gourdon & Patrick Demichel
|-
| December 8, 2006 || 116,580,041 || Alexander J. Yee
|}

References

1. Krämer 2005
2. Havil, page 97
3.
4. Havil, 117-118
5. Gourdon, X., Sebah, P; The Euler constant: g


★ Computational Strategies for the Riemann Zeta Function, Jonathan M. Borwein, David M. Bradley, Richard E. Crandall, , , Journal of Computational and Applied Mathematics, 2000 ''(Provides a derivation of the sums over Riemann zeta)''

★ Knuth, Donald E., ''The Art of Computer Programming'', volume 1, Addison-Wesley. 1997 (third edition). ISBN 0-201-89683-4

★ Gamma: Exploring Euler's Constant, , Julian, Havil, Princeton University Press, 2003, ISBN 0-691-09983-9

★

★ ''Euler-Mascheroni Constant'' from the Mathcad Library

★ Simon Plouffe, Value of γ to 10 million decimal places

★ Jonathan Sondow, "An antisymmetric formula for Euler's constant," Mathematics Magazine 71 (1998) 219-220.

★ Jonathan Sondow, "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant" (2002, preprint) with an Appendix by Sergey Zlobin.

★ Jonathan Sondow, "An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma" (2003, preprint).

★ Jonathan Sondow, "Criteria for irrationality of Euler's constant," Proceedings of the American Mathematical Society 131 (2003) 3335-3344.

★ Jonathan Sondow, "Double integrals for Euler's constant and ln 4/pi and an analog of Hadjicostas's formula," American Mathematical Monthly 112 (2005) 61-65.

★ Jonathan Sondow, "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/pi" (2005, preprint).

★ Jonathan Sondow and Wadim Zudilin, "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper," Ramanujan J. (to appear).

★ Stefan Krämer. Die Eulersche Konstante γ und verwandte Zahlen. Diplomarbeit, Universität Göttingen, 2005.

★ Stefan Krämer. Euler's Constant γ=0.577... Its Mathematics and History