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The area of the blue region converges to the Euler–Mascheroni constant.

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).

It is defined as the limiting difference between the harmonic series and the natural logarithm:

γ=limn→∞(−ln⁡n+∑k=1n1k)=∫1∞(−1x+1⌊x⌋)dx.{displaystyle {begin{aligned}gamma &=lim _{nto infty }left(-ln n+sum _{k=1}^{n}{frac {1}{k}}right)\[5px]&=int _{1}^{infty }left(-{frac {1}{x}}+{frac {1}{lfloor xrfloor }}right),dx.end{aligned}}}

Here, ⌊x⌋{displaystyle lfloor xrfloor } represents the floor function.

The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:

0.57721566490153286060651209008240243104215933593992... (sequence A001620 in the OEIS)

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.^[2] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835^[3] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.^[4]

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 Laurent 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
  


• An upper bound on Shannon entropy in quantum information theory^[5]
  

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.^[6] 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).{displaystyle -gamma =Gamma '(1)=Psi (1).}

This is equal to the limits:

−γ=limz→0(Γ(z)−1z)=limz→0(Ψ(z)+1z).{displaystyle {begin{aligned}-gamma &=lim _{zto 0}left(Gamma (z)-{frac {1}{z}}right)\&=lim _{zto 0}left(Psi (z)+{frac {1}{z}}right).end{aligned}}}

Further limit results are (Krämer, 2005):

limz→01z(1Γ(1+z)−1Γ(1−z))=2γlimz→01z(1Ψ(1−z)−1Ψ(1+z))=π23γ2.{displaystyle {begin{aligned}lim _{zto 0}{frac {1}{z}}left({frac {1}{Gamma (1+z)}}-{frac {1}{Gamma (1-z)}}right)&=2gamma \lim _{zto 0}{frac {1}{z}}left({frac {1}{Psi (1-z)}}-{frac {1}{Psi (1+z)}}right)&={frac {pi ^{2}}{3gamma ^{2}}}.end{aligned}}}

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

γ=limn→∞(Γ(1n)Γ(n+1)n1+1nΓ(2+n+1n)−n2n+1)=limm→∞∑k=1m(mk)(−1)kkln⁡(Γ(k+1)).{displaystyle {begin{aligned}gamma &=lim _{nto infty }left({frac {Gamma left({frac {1}{n}}right)Gamma (n+1),n^{1+{frac {1}{n}}}}{Gamma left(2+n+{frac {1}{n}}right)}}-{frac {n^{2}}{n+1}}right)\&=lim limits _{mto infty }sum _{k=1}^{m}{m choose k}{frac {(-1)^{k}}{k}}ln {big (}Gamma (k+1){big )}.end{aligned}}}

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)mζ(m)m=ln⁡4π+∑m=2∞(−1)mζ(m)2m−1m.{displaystyle {begin{aligned}gamma &=sum _{m=2}^{infty }(-1)^{m}{frac {zeta (m)}{m}}\&=ln {frac {4}{pi }}+sum _{m=2}^{infty }(-1)^{m}{frac {zeta (m)}{2^{m-1}m}}.end{aligned}}}

Other series related to the zeta function include:

γ=32−ln⁡2−∑m=2∞(−1)mm−1m(ζ(m)−1)=limn→∞(2n−12n−ln⁡n+∑k=2n(1k−ζ(1−k)nk))=limn→∞(2ne2n∑m=0∞2mn(m+1)!∑t=0m1t+1−nln⁡2+O(12ne2n)).{displaystyle {begin{aligned}gamma &={tfrac {3}{2}}-ln 2-sum _{m=2}^{infty }(-1)^{m},{frac {m-1}{m}}{big (}zeta (m)-1{big )}\&=lim _{nto infty }left({frac {2n-1}{2n}}-ln n+sum _{k=2}^{n}left({frac {1}{k}}-{frac {zeta (1-k)}{n^{k}}}right)right)\&=lim _{nto infty }left({frac {2^{n}}{e^{2^{n}}}}sum _{m=0}^{infty }{frac {2^{mn}}{(m+1)!}}sum _{t=0}^{m}{frac {1}{t+1}}-nln 2+Oleft({frac {1}{2^{n},e^{2^{n}}}}right)right).end{aligned}}}

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=lims→1+∑n=1∞(1ns−1sn)=lims→1(ζ(s)−1s−1)=lims→0ζ(1+s)+ζ(1−s)2{displaystyle {begin{aligned}gamma &=lim _{sto 1^{+}}sum _{n=1}^{infty }left({frac {1}{n^{s}}}-{frac {1}{s^{n}}}right)\&=lim _{sto 1}left(zeta (s)-{frac {1}{s-1}}right)\&=lim _{sto 0}{frac {zeta (1+s)+zeta (1-s)}{2}}end{aligned}}}

and de la Vallée-Poussin's formula

γ=limn→∞1n∑k=1n(⌈nk⌉−nk){displaystyle gamma =lim _{nto infty }{frac {1}{n}},sum _{k=1}^{n}left(leftlceil {frac {n}{k}}rightrceil -{frac {n}{k}}right)}

where ⌈⌉{displaystyle lceil ,rceil } 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=1n1k−ln⁡n−∑m=2∞ζ(m,n+1)m,{displaystyle gamma =sum _{k=1}^{n}{frac {1}{k}}-ln n-sum _{m=2}^{infty }{frac {zeta (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:

Hn=ln⁡(n)+γ+12n−112n2+1120n4−ε,{displaystyle H_{n}=ln(n)+gamma +{frac {1}{2n}}-{frac {1}{12n^{2}}}+{frac {1}{120n^{4}}}-varepsilon ,}

where 0 < ε < 1/252n⁶.

Integrals

γ equals the value of a number of definite integrals:

γ=−∫0∞e−xln⁡xdx=−∫01ln⁡(ln⁡1x)dx=∫0∞(1ex−1−1x⋅ex)dx=∫01(1ln⁡x+11−x)dx=∫0∞(11+xk−e−x)dxx,k>0=∫01Hxdx,{displaystyle {begin{aligned}gamma &=-int _{0}^{infty }e^{-x}ln x,dx\&=-int _{0}^{1}ln left(ln {frac {1}{x}}right)dx\&=int _{0}^{infty }left({frac {1}{e^{x}-1}}-{frac {1}{xcdot e^{x}}}right)dx\&=int _{0}^{1}left({frac {1}{ln x}}+{frac {1}{1-x}}right)dx\&=int _{0}^{infty }left({frac {1}{1+x^{k}}}-e^{-x}right){frac {dx}{x}},quad k>0\&=int _{0}^{1}H_{x},dx,end{aligned}}}

where H_x is the fractional harmonic number.

Definite integrals in which γ appears include:

∫0∞e−x2ln⁡xdx=−(γ+2ln⁡2)π4∫0∞e−xln2⁡xdx=γ2+π26.{displaystyle {begin{aligned}int _{0}^{infty }e^{-x^{2}}ln x,dx&=-{frac {(gamma +2ln 2){sqrt {pi }}}{4}}\int _{0}^{infty }e^{-x}ln ^{2}x,dx&=gamma ^{2}+{frac {pi ^{2}}{6}}.end{aligned}}}

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

γ=∫01∫01x−1(1−xy)ln⁡xydxdy=∑n=1∞(1n−ln⁡n+1n).{displaystyle {begin{aligned}gamma &=int _{0}^{1}int _{0}^{1}{frac {x-1}{(1-xy)ln xy}},dx,dy\&=sum _{n=1}^{infty }left({frac {1}{n}}-ln {frac {n+1}{n}}right).end{aligned}}}

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

ln⁡4π=∫01∫01x−1(1+xy)ln⁡xydxdy=∑n=1∞((−1)n−1(1n−ln⁡n+1n)).{displaystyle {begin{aligned}ln {frac {4}{pi }}&=int _{0}^{1}int _{0}^{1}{frac {x-1}{(1+xy)ln xy}},dx,dy\&=sum _{n=1}^{infty }left((-1)^{n-1}left({frac {1}{n}}-ln {frac {n+1}{n}}right)right).end{aligned}}}

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∞N1(n)+N0(n)2n(2n+1)ln⁡4π=∑n=1∞N1(n)−N0(n)2n(2n+1),{displaystyle {begin{aligned}gamma &=sum _{n=1}^{infty }{frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}\ln {frac {4}{pi }}&=sum _{n=1}^{infty }{frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}},end{aligned}}}

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=∫01(11+x∑n=1∞x2n−1)dx.{displaystyle gamma =int _{0}^{1}left({frac {1}{1+x}}sum _{n=1}^{infty }x^{2^{n}-1}right),dx.}

Series expansions

In general,

γ=limn→∞(11+12+13+…+1n−ln⁡(n+α))≡limn→∞γn(α){displaystyle gamma =lim _{nto infty }left({frac {1}{1}}+{frac {1}{2}}+{frac {1}{3}}+ldots +{frac {1}{n}}-ln(n+alpha )right)equiv lim _{nto infty }gamma _{n}(alpha )}

for any α>−n{displaystyle alpha >-n}. However, the rate of convergence of this expansion depends significantly on α{displaystyle alpha }. In particular, γn(1/2){displaystyle gamma _{n}(1/2)} exhibits much more rapid convergence than the conventional expansion γn(0){displaystyle gamma _{n}(0)}.^[7][8] This is because

12(n+1)<γn(0)−γ<12n,{displaystyle {frac {1}{2(n+1)}}<gamma _{n}(0)-gamma <{frac {1}{2n}},}

while

124(n+1)2<γn(1/2)−γ<124n2.{displaystyle {frac {1}{24(n+1)^{2}}}<gamma _{n}(1/2)-gamma <{frac {1}{24n^{2}}}.}

Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.

Euler showed that the following infinite series approaches γ:

γ=∑k=1∞(1k−ln⁡(1+1k)).{displaystyle gamma =sum _{k=1}^{infty }left({frac {1}{k}}-ln left(1+{frac {1}{k}}right)right).}

The series for γ is equivalent to a series Nielsen found in 1897:^[9]

γ=1−∑k=2∞(−1)k⌊log2⁡k⌋k+1.{displaystyle gamma =1-sum _{k=2}^{infty }(-1)^{k}{frac {leftlfloor log _{2}krightrfloor }{k+1}}.}

In 1910, Vacca found the closely related series^[10]

γ=∑k=2∞(−1)k⌊log2⁡k⌋k=12−13+2(14−15+16−17)+3(18−19+110−111+⋯−115)+⋯,{displaystyle {begin{aligned}gamma &=sum _{k=2}^{infty }(-1)^{k}{frac {leftlfloor log _{2}krightrfloor }{k}}\[5pt]&={tfrac {1}{2}}-{tfrac {1}{3}}+2left({tfrac {1}{4}}-{tfrac {1}{5}}+{tfrac {1}{6}}-{tfrac {1}{7}}right)+3left({tfrac {1}{8}}-{tfrac {1}{9}}+{tfrac {1}{10}}-{tfrac {1}{11}}+cdots -{tfrac {1}{15}}right)+cdots ,end{aligned}}}

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⌋2−1k)=∑k=2∞k−⌊k⌋2k⌊k⌋2=12+23+122∑k=12⋅2kk+22+132∑k=13⋅2kk+32+⋯{displaystyle {begin{aligned}gamma +zeta (2)&=sum _{k=2}^{infty }left({frac {1}{leftlfloor {sqrt {k}}rightrfloor ^{2}}}-{frac {1}{k}}right)\[5pt]&=sum _{k=2}^{infty }{frac {k-leftlfloor {sqrt {k}}rightrfloor ^{2}}{kleftlfloor {sqrt {k}}rightrfloor ^{2}}}\[5pt]&={frac {1}{2}}+{frac {2}{3}}+{frac {1}{2^{2}}}sum _{k=1}^{2cdot 2}{frac {k}{k+2^{2}}}+{frac {1}{3^{2}}}sum _{k=1}^{3cdot 2}{frac {k}{k+3^{2}}}+cdots end{aligned}}}

From the Malmsten–Kummer expansion for the logarithm of the gamma function^[11] we get:

γ=ln⁡π−4ln⁡(Γ(34))+4π∑k=1∞(−1)k+1ln⁡(2k+1)2k+1.{displaystyle gamma =ln pi -4ln left(Gamma ({tfrac {3}{4}})right)+{frac {4}{pi }}sum _{k=1}^{infty }(-1)^{k+1}{frac {ln(2k+1)}{2k+1}}.}

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

γ=∑n=1∞|Gn|n=12+124+172+192880+3800+⋯,{displaystyle gamma =sum _{n=1}^{infty }{frac {|G_{n}|}{n}}={frac {1}{2}}+{frac {1}{24}}+{frac {1}{72}}+{frac {19}{2880}}+{frac {3}{800}}+cdots ,}

where G_n are Gregory coefficients.^[12]

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

Hn=γ+ln⁡n+12n−∑k=2∞(k−1)!|Gk|n(n+1)⋯(n+k−1),n=1,2,…,=γ+ln⁡n+12n−112n(n+1)−112n(n+1)(n+2)−19120n(n+1)(n+2)(n+3)−⋯{displaystyle {begin{aligned}H_{n}&=gamma +ln n+{frac {1}{2n}}-sum _{k=2}^{infty }{frac {(k-1)!|G_{k}|}{n(n+1)cdots (n+k-1)}},&&n=1,2,ldots ,\&=gamma +ln n+{frac {1}{2n}}-{frac {1}{12n(n+1)}}-{frac {1}{12n(n+1)(n+2)}}-{frac {19}{120n(n+1)(n+2)(n+3)}}-cdots &&end{aligned}}}

and is convergent for all n.

A similar series with the Cauchy numbers of the second kind C_n is^[13]

γ=1−∑n=1∞Cnn(n+1)!=1−14−572−132−25114400−191728−…{displaystyle gamma =1-sum _{n=1}^{infty }{frac {C_{n}}{n,(n+1)!}}=1-{frac {1}{4}}-{frac {5}{72}}-{frac {1}{32}}-{frac {251}{14400}}-{frac {19}{1728}}-ldots }

Blagouchine (2018) found an interesting generalisation of the Fontana-Mascheroni series

γ=∑n=1∞(−1)n+12n{ψn(a)+ψn(−a1+a)},a>−1{displaystyle gamma =sum _{n=1}^{infty }{frac {(-1)^{n+1}}{2n}}{Big {}psi _{n}(a)+psi _{n}{Big (}-{frac {a}{1+a}}{Big )}{Big }},quad a>-1}

where ψ_n(a) are the Bernoulli polynomials of the second kind, which are defined by the generating function

z(1+z)sln⁡(1+z)=∑n=0∞znψn(s),|z|<1,{displaystyle {frac {z(1+z)^{s}}{ln(1+z)}}=sum _{n=0}^{infty }z^{n}psi _{n}(s),qquad |z|<1,}

For any rational a this series contains rational terms only. For example, at a = 1, it becomes

γ=34−1196−172−31146080−51152−72912322432−243100352−…{displaystyle gamma ={frac {3}{4}}-{frac {11}{96}}-{frac {1}{72}}-{frac {311}{46080}}-{frac {5}{1152}}-{frac {7291}{2322432}}-{frac {243}{100352}}-ldots }

see OEIS OEIS: A302120 and OEIS: A302121. Other series with the same polynomials include these examples:

γ=−ln⁡(a+1)−∑n=1∞(−1)nψn(a)n,ℜ(a)>−1{displaystyle gamma =-ln(a+1)-sum _{n=1}^{infty }{frac {(-1)^{n}psi _{n}(a)}{n}},qquad Re (a)>-1}

and

γ=−21+2a{ln⁡Γ(a+1)−12ln⁡(2π)+12+∑n=1∞(−1)nψn+1(a)n},ℜ(a)>−1{displaystyle gamma =-{frac {2}{1+2a}}left{ln Gamma (a+1)-{frac {1}{2}}ln(2pi )+{frac {1}{2}}+sum _{n=1}^{infty }{frac {(-1)^{n}psi _{n+1}(a)}{n}}right},qquad Re (a)>-1}

where Γ(a) is the gamma function.^[14]

A series related to the Akiyama-Tanigawa algorithm is

γ=ln⁡(2π)−2−2∑n=1∞(−1)nGn(2)n=ln⁡(2π)−2+23+124+7540+172880+4112600+…{displaystyle gamma =ln(2pi )-2-2sum _{n=1}^{infty }{frac {(-1)^{n}G_{n}(2)}{n}}=ln(2pi )-2+{frac {2}{3}}+{frac {1}{24}}+{frac {7}{540}}+{frac {17}{2880}}+{frac {41}{12600}}+ldots }

where G_n(2) are the Gregory coefficients of the second order.^[14]

Series of prime numbers:

γ=limn→∞(ln⁡n−∑p≤nln⁡pp−1).{displaystyle gamma =lim _{nto infty }left(ln n-sum _{pleq n}{frac {ln p}{p-1}}right).}

Series relating to square roots:^[15]

γ=limn→∞(∑k=1n1k−ln⁡∑k=1nk)−ln⁡22.{displaystyle gamma =lim _{nto infty }left(sum _{k=1}^{n}{frac {1}{k}}-ln {sqrt {sum _{k=1}^{n}k}},right)-{frac {ln 2}{2}}.}

Asymptotic expansions

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

γ∼Hn−ln⁡n−12n+112n2−1120n4+⋯{displaystyle gamma sim H_{n}-ln n-{frac {1}{2n}}+{frac {1}{12n^{2}}}-{frac {1}{120n^{4}}}+cdots } (Euler)

γ∼Hn−ln⁡(n+12+124n−148n3+⋯){displaystyle gamma sim H_{n}-ln left({n+{frac {1}{2}}+{frac {1}{24n}}-{frac {1}{48n^{3}}}+cdots }right)} (Negoi)

γ∼Hn−ln⁡n+ln⁡(n+1)2−16n(n+1)+130n2(n+1)2−⋯{displaystyle gamma sim H_{n}-{frac {ln n+ln(n+1)}{2}}-{frac {1}{6n(n+1)}}+{frac {1}{30n^{2}(n+1)^{2}}}-cdots } (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:

eγ=limn→∞1ln⁡pn∏i=1npipi−1.{displaystyle e^{gamma }=lim _{nto infty }{frac {1}{ln p_{n}}}prod _{i=1}^{n}{frac {p_{i}}{p_{i}-1}}.}

This restates the third of Mertens' theorems.^[16] The numerical value of e^γ is:

1.78107241799019798523650410310717954916964521430343... OEIS: A073004.

Other infinite products relating to e^γ include:

e1+γ22π=∏n=1∞e−1+12n(1+1n)ne3+2γ2π=∏n=1∞e−2+2n(1+2n)n.{displaystyle {begin{aligned}{frac {e^{1+{frac {gamma }{2}}}}{sqrt {2pi }}}&=prod _{n=1}^{infty }e^{-1+{frac {1}{2n}}}left(1+{frac {1}{n}}right)^{n}\{frac {e^{3+2gamma }}{2pi }}&=prod _{n=1}^{infty }e^{-2+{frac {2}{n}}}left(1+{frac {2}{n}}right)^{n}.end{aligned}}}

These products result from the Barnes G-function.

We also have

eγ=21⋅221⋅33⋅23⋅41⋅334⋅24⋅441⋅36⋅55⋯{displaystyle e^{gamma }={sqrt {frac {2}{1}}}cdot {sqrt[{3}]{frac {2^{2}}{1cdot 3}}}cdot {sqrt[{4}]{frac {2^{3}cdot 4}{1cdot 3^{3}}}}cdot {sqrt[{5}]{frac {2^{4}cdot 4^{4}}{1cdot 3^{6}cdot 5}}}cdots }

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

∏k=0n(k+1)(−1)k+1(nk).{displaystyle prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n choose 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, ...] OEIS: A002852, which has no apparent pattern. The continued fraction is known to have at least 470,000 terms,^[6] and it has infinitely many terms if and only if γ is irrational.

Generalizations

abm(x) = γ_−x

Euler's generalized constants are given by

γα=limn→∞(∑k=1n1kα−∫1n1xαdx),{displaystyle gamma _{alpha }=lim _{nto infty }left(sum _{k=1}^{n}{frac {1}{k^{alpha }}}-int _{1}^{n}{frac {1}{x^{alpha }}},dxright),}

for 0 < α < 1, with γ as the special case α = 1.^[17] This can be further generalized to

cf=limn→∞(∑k=1nf(k)−∫1nf(x)dx){displaystyle c_{f}=lim _{nto infty }left(sum _{k=1}^{n}f(k)-int _{1}^{n}f(x),dxright)}

for some arbitrary decreasing function f. For example,

fn(x)=(ln⁡x)nx{displaystyle f_{n}(x)={frac {(ln x)^{n}}{x}}}

gives rise to the Stieltjes constants, and

fa(x)=x−a{displaystyle f_{a}(x)=x^{-a}}

gives

γfa=(a−1)ζ(a)−1a−1{displaystyle gamma _{f_{a}}={frac {(a-1)zeta (a)-1}{a-1}}}

where again the limit

γ=lima→1(ζ(a)−1a−1){displaystyle gamma =lim _{ato 1}left(zeta (a)-{frac {1}{a-1}}right)}

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:^[18][19]

γ(a,q)=limx→∞(∑0<n≤xn≡a(modq)1n−ln⁡xq).{displaystyle gamma (a,q)=lim _{xto infty }left(sum _{0<nleq x atop nequiv a{pmod {q}}}{frac {1}{n}}-{frac {ln x}{q}}right).}

The basic properties are

γ(0,q)=γ−ln⁡qq,∑a=0q−1γ(a,q)=γ,qγ(a,q)=γ−∑j=1q−1e−2πaijqln⁡(1−e2πijq),{displaystyle {begin{aligned}gamma (0,q)&={frac {gamma -ln q}{q}},\sum _{a=0}^{q-1}gamma (a,q)&=gamma ,\qgamma (a,q)&=gamma -sum _{j=1}^{q-1}e^{-{frac {2pi aij}{q}}}ln left(1-e^{frac {2pi ij}{q}}right),end{aligned}}}

and if gcd(a,q) = d then

qγ(a,q)=qdγ(ad,qd)−ln⁡d.{displaystyle qgamma (a,q)={frac {q}{d}}gamma left({frac {a}{d}},{frac {q}{d}}right)-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 20th–22nd and 31st-32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.

Notes

Footnotes

References

• 
  Alabdulmohsin, Ibrahim M. (2018), Summability Calculus. A Comprehensive Theory of Fractional Finite Sums, Springer-Verlag, ISBN 9783319746487
  


• 
  Blagouchine, Iaroslav V. (2014), "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results", The Ramanujan Journal, 35 (1): 21–110, doi:10.1007/s11139-013-9528-5
  PDF
  


• 
  Blagouchine, Iaroslav V. (2016), "Expansions of generalized Euler's constants into the series of polynomials in π⁻² and into the formal enveloping series with rational coefficients only", J. Number Theory, 158: 365–396, arXiv:1501.00740, doi:10.1016/j.jnt.2015.06.012
  


• 
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  Borwein, Jonathan M.; David M. Bradley; Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function" (PDF). Journal of Computational and Applied Mathematics. 121 (1–2): 11. Bibcode:2000JCoAM.121..247B. doi:10.1016/s0377-0427(00)00336-8. Derives γ as sums over Riemann zeta functions.


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  Carl Anton, Bretschneider (1837). "Theoriae logarithmi integralis lineamenta nova". Crelle's Journal. 17: 257–285. (submitted 1835)


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  James Whitbread Lee Glaisher (1910). "On Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 41: 365–368.
  


• Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ."


• Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: γ."


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  Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 978-0-691-09983-5.
  


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  Hardy, G. H. (1912). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
  


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  Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (44): 339–360.
  


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  Karatsuba, E.A. (2000). "On the computation of the Euler constant γ". Journal of Numerical Algorithms. 24 (1–2): 83–97.
  


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  Kluyver, J.C. (1927). "On certain series of Mr. Hardy". Q. J. Pure Appl. Math. 50: 185–192.
  


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  Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley.
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• Krämer, Stefan (2005) Die Eulersche Konstante γ und verwandte Zahlen. Ph.D. Thesis, University of Göttingen, Germany.


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  Lagarias, Jeffrey C. (October 2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 556. arXiv:1303.1856. Bibcode:1994BAMaS..30..205W. doi:10.1090/s0273-0979-2013-01423-x.
  


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  Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
  


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  Lorenzo Mascheroni (1790). "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur". Galeati, Ticini.


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


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  Sondow, Jonathan (2002). "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant". Mathematica Slovaca. 59: 307–314. arXiv:math.NT/0211075. Bibcode:2002math.....11075S. with an Appendix by Sergey Zlobin
  


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  Sondow, Jonathan (2003). "An infinite product for e^γ via hypergeometric formulas for Euler's constant, γ". arXiv:math.CA/0306008.
  


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  Sondow, Jonathan (2003). "Criteria for irrationality of Euler's constant". Proceedings of the American Mathematical Society. 131. pp. 3335–3344. arXiv:math.NT/0209070.
  


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  Sondow, Jonathan (2005). "Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula". American Mathematical Monthly. 112 (1): 61–65. arXiv:math.CA/0211148. doi:10.2307/30037385. JSTOR 30037385.
  


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  Sondow, Jonathan (2005). "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π". arXiv:math.NT/0508042.
  


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  Sondow, Jonathan; Zudilin, Wadim (2006). "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper". The Ramanujan Journal. 12 (2): 225–244. arXiv:math.NT/0304021. doi:10.1007/s11139-006-0075-1. Ramanujan Journal 12: 225-244.


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  Vacca, G. (1926). "Nuova serie per la costante di Eulero, C = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche". Matematiche e Naturali. 6 (3): 19–20.
  

External links

• Weisstein, Eric W. "Euler–Mascheroni constant". MathWorld.


• Jonathan Sondow.


• 
  Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)


• Further formulae which make use of the constant: Gourdon and Sebah (2004).