Euler–Mascheroni constant - Alchetron, the free social encyclopedia

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (γ).

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.

Appearances

The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):

Expressions involving the exponential integral*

The Laplace transform* of the natural logarithm

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

Calculations of the digamma function

A product formula for the gamma function

An inequality for Euler's totient function

The growth rate of the divisor function

In Dimensional regularization of Feynman diagrams in Quantum Field Theory

The calculation of the Meissel–Mertens constant

The third of Mertens' theorems*

Solution of the second kind to Bessel's equation

In the regularization/renormalization of the Harmonic series as a finite value

The mean of the Gumbel distribution

The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.

The answer to the coupon collector's problem*

In some formulations of Zipf's law

A definition of the cosine integral*

Lower bounds to a prime gap.

Properties

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Continued fraction analysis reveals that if γ is rational, its denominator must be greater than 10²⁴²⁰⁸⁰. The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics. Also see Sondow (2003a).

Relation to gamma function

γ is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:

− γ = Γ ′ ( 1 ) = Ψ ( 1 ) .

This is equal to the limits:

− γ = lim z → 0 ( Γ ( z ) − 1 z ) = lim z → 0 ( Ψ ( z ) + 1 z ) .

Further limit results are (Krämer, 2005):

lim z → 0 1 z ( 1 Γ ( 1 + z ) − 1 Γ ( 1 − z ) ) = 2 γ lim z → 0 1 z ( 1 Ψ ( 1 − z ) − 1 Ψ ( 1 + z ) ) = π 2 3 γ 2 .

A limit related to the beta function (expressed in terms of gamma functions) is

γ = lim n → ∞ ( Γ ( 1 n ) Γ ( n + 1 ) n 1 + 1 n Γ ( 2 + n + 1 n ) − n 2 n + 1 ) = lim m → ∞ ∑ k = 1 m ( m k ) ( − 1 ) k k ln ⁡ ( Γ ( k + 1 ) ) .

Relation to the zeta function

γ can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:

γ = ∑ m = 2 ∞ ( − 1 ) m ζ ( m ) m = ln ⁡ 4 π + ∑ m = 2 ∞ ( − 1 ) m ζ ( m ) 2 m − 1 m .

Other series related to the zeta function include:

γ = 3 2 − ln ⁡ 2 − ∑ m = 2 ∞ ( − 1 ) m m − 1 m ( ζ ( m ) − 1 ) = lim n → ∞ ( 2 n − 1 2 n − ln ⁡ n + ∑ k = 2 n ( 1 k − ζ ( 1 − k ) n k ) ) = lim n → ∞ ( 2 n e 2 n ∑ m = 0 ∞ 2 m n ( m + 1 ) ! ∑ t = 0 m 1 t + 1 − n ln ⁡ 2 + O ( 1 2 n e 2 n ) ) .

The error term in the last equation is a rapidly decreasing function of n. As a result, the formula is well-suited for efficient computation of the constant to high precision.

Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit (Sondow, 1998):

γ = lim s → 1 + ∑ n = 1 ∞ ( 1 n s − 1 s n ) = lim s → 1 ( ζ ( s ) − 1 s − 1 ) = lim s → 0 ζ ( 1 + s ) + ζ ( 1 − s ) 2

and de la Vallée-Poussin's formula

γ = lim n → ∞ 1 n ∑ k = 1 n ( ⌈ n k ⌉ − n k )

where ⌈ ⌉ are ceiling brackets.

Closely related to this is the rational zeta series expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:

γ = ∑ k = 1 n 1 k − ln ⁡ n − ∑ m = 2 ∞ ζ ( m , n + 1 ) m ,

where ζ(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 = ln ⁡ ( n ) + γ + 1 2 n − 1 12 n 2 + 1 120 n 4 − ε ,

where 0 < ε < 1/252n⁶.

Integrals

γ equals the value of a number of definite integrals:

γ = − ∫ 0 ∞ e − x ln ⁡ x d x = − ∫ 0 1 ln ⁡ ( ln ⁡ 1 x ) d x = ∫ 0 ∞ ( 1 e x − 1 − 1 x ⋅ e x ) d x = ∫ 0 1 ( 1 ln ⁡ x + 1 1 − x ) d x = ∫ 0 ∞ ( 1 1 + x k − e − x ) d x x , k > 0 = ∫ 0 1 H x d x ,

where H_x is the fractional harmonic number.

Definite integrals in which γ appears include:

∫ 0 ∞ e − x 2 ln ⁡ x d x = − ( γ + 2 ln ⁡ 2 ) π 4 ∫ 0 ∞ e − x ln 2 ⁡ x d x = γ 2 + π 2 6 .

One can express γ using a special case of Hadjicostas's formula as a double integral (Sondow 2003a, 2005) with equivalent series:

γ = ∫ 0 1 ∫ 0 1 x − 1 ( 1 − x y ) ln ⁡ x y d x d y = ∑ n = 1 ∞ ( 1 n − ln ⁡ n + 1 n ) .

An interesting comparison by J. Sondow (2005) is the double integral and alternating series

ln ⁡ 4 π = ∫ 0 1 ∫ 0 1 x − 1 ( 1 + x y ) ln ⁡ x y d x d y = ∑ n = 1 ∞ ( ( − 1 ) n − 1 ( 1 n − ln ⁡ n + 1 n ) ) .

It shows that ln 4/π may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series (see Sondow 2005 #2)

γ = ∑ n = 1 ∞ N 1 ( n ) + N 0 ( n ) 2 n ( 2 n + 1 ) ln ⁡ 4 π = ∑ n = 1 ∞ N 1 ( n ) − N 0 ( n ) 2 n ( 2 n + 1 ) ,

where N₁(n) and N₀(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.

We have also Catalan's 1875 integral (see Sondow and Zudilin)

γ = ∫ 0 1 ( 1 1 + x ∑ n = 1 ∞ x 2 n − 1 ) d x .

Series expansions

Euler showed that the following infinite series approaches γ:

γ = ∑ k = 1 ∞ ( 1 k − ln ⁡ ( 1 + 1 k ) ) .

The series for γ is equivalent to a series Nielsen found in 1897:

γ = 1 − ∑ k = 2 ∞ ( − 1 ) k ⌊ log 2 ⁡ k ⌋ k + 1 .

In 1910, Vacca found the closely related series:

γ = ∑ k = 2 ∞ ( − 1 ) k ⌊ log 2 ⁡ k ⌋ k = 1 2 − 1 3 + 2 ( 1 4 − 1 5 + 1 6 − 1 7 ) + 3 ( 1 8 − 1 9 + 1 10 − 1 11 + ⋯ − 1 15 ) + ⋯ ,

where log₂ is the logarithm to base 2 and ⌊ ⌋ is the floor function.

In 1926 he found a second series:

γ + ζ ( 2 ) = ∑ k = 2 ∞ ( 1 ⌊ k ⌋ 2 − 1 k ) = ∑ k = 2 ∞ k − ⌊ k ⌋ 2 k ⌊ k ⌋ 2 = 1 2 + 2 3 + 1 2 2 ∑ k = 1 2 ⋅ 2 k k + 2 2 + 1 3 2 ∑ k = 1 3 ⋅ 2 k k + 3 2 + ⋯

From the Malmsten–Kummer expansion for the logarithm of the gamma function we get:

γ = ln ⁡ π − 4 ln ⁡ ( Γ ( 3 4 ) ) + 4 π ∑ k = 1 ∞ ( − 1 ) k + 1 ln ⁡ ( 2 k + 1 ) 2 k + 1 .

An important expansion for Euler's constant is due to Fontana and Mascheroni

γ = ∑ n = 1 ∞ | G n | n = 1 2 + 1 24 + 1 72 + 19 2880 + 3 800 + ⋯ ,

where G_n are Gregory coefficients.

Another important expansion with the Gregory coefficients involving Euler's constant is:

H n = γ + ln ⁡ n + 1 2 n − ∑ k = 2 ∞ ( k − 1 ) ! | G k | n ( n + 1 ) ⋯ ( n + k − 1 ) , n = 1 , 2 , … , = γ + ln ⁡ n + 1 2 n − 1 12 n ( n + 1 ) − 1 12 n ( n + 1 ) ( n + 2 ) − 19 120 n ( n + 1 ) ( n + 2 ) ( n + 3 ) − ⋯

and is convergent for all n.

Series of prime numbers:

γ = lim n → ∞ ( ln ⁡ n − ∑ p ≤ n ln ⁡ p p − 1 ) .

Series relating to square roots:

γ = lim n → ∞ ( ∑ k = 1 n 1 k − ln ⁡ ∑ k = 1 n k ) − ln ⁡ 2 2 .

Asymptotic expansions

γ equals the following asymptotic formulas (where H_n is the nth harmonic number):

γ ∼ H n − ln ⁡ n − 1 2 n + 1 12 n 2 − 1 120 n 4 + ⋯ (Euler) γ ∼ H n − ln ⁡ ( n + 1 2 + 1 24 n − 1 48 n 3 + ⋯ ) (Negoi) γ ∼ H n − ln ⁡ n + ln ⁡ ( n + 1 ) 2 − 1 6 n ( n + 1 ) + 1 30 n 2 ( n + 1 ) 2 − ⋯ (Cesàro)

The third formula is also called the Ramanujan expansion.

Exponential

The constant e^γ is important in number theory. Some authors denote this quantity simply as γ′. e^γ equals the following limit, where p_n is the nth prime number:

e γ = lim n → ∞ 1 ln ⁡ p n ∏ i = 1 n p i p i − 1 .

This restates the third of Mertens' theorems. The numerical value of e^γ is:

1.78107241799019798523650410310717954916964521430343… A073004.

Other infinite products relating to e^γ include:

e 1 + γ 2 2 π = ∏ n = 1 ∞ e − 1 + 1 2 n ( 1 + 1 n ) n e 3 + 2 γ 2 π = ∏ n = 1 ∞ e − 2 + 2 n ( 1 + 2 n ) n .

These products result from the Barnes G-function.

We also have

e γ = 2 1 ⋅ 2 2 1 ⋅ 3 3 ⋅ 2 3 ⋅ 4 1 ⋅ 3 3 4 ⋅ 2 4 ⋅ 4 4 1 ⋅ 3 6 ⋅ 5 5 ⋯

where the nth factor is the (n + 1)th root of

∏ k = 0 n ( k + 1 ) ( − 1 ) k + 1 ( n k ) .

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.

Continued fraction

The continued fraction expansion of γ is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] A002852, of which there is no apparent pattern. The continued fraction is known to have at least 470,000 terms, and it has infinitely many terms if and only if γ is irrational.

Generalizations

Euler's generalized constants are given by

γ α = lim n → ∞ ( ∑ k = 1 n 1 k α − ∫ 1 n 1 x α d x ) ,

for 0 < α < 1, with γ as the special case α = 1. This can be further generalized to

c f = lim n → ∞ ( ∑ k = 1 n f ( k ) − ∫ 1 n f ( x ) d x )

for some arbitrary decreasing function f. For example,

f n ( x ) = ln n ⁡ ( x ) x

gives rise to the Stieltjes constants, and

f a ( x ) = x − a

gives

γ f a = ( a − 1 ) ζ ( a ) − 1 a − 1

where again the limit

γ = lim a → 1 ( ζ ( a ) − 1 a − 1 )

appears.

A two-dimensional limit generalization is the Masser–Gramain constant.

Euler–Lehmer constants are given by summation of inverses of numbers in a common modulo class:

γ ( a , q ) = lim x → ∞ ( ∑ 0 < n ≤ x n ≡ a ( mod q ) 1 n − ln ⁡ x q ) .

The basic properties are

γ ( 0 , q ) = γ − ln ⁡ q q , ∑ a = 0 q − 1 γ ( a , q ) = γ , q γ ( a , q ) = γ − ∑ j = 1 q − 1 e − 2 π a i j q ln ⁡ ( 1 − e 2 π i j q ) ,

and if gcd(a,q) = d then

q γ ( a , q ) = q d γ ( a d , q d ) − ln ⁡ d .

Published digits

Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 19th–21st and 32nd decimal places; starting from the 19th digit, he calculated …1811209008239 when the correct value is …0651209008240.

References

Euler–Mascheroni constant Wikipedia

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