Euler-Mascheroni Constant
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Euler's constant (sometimes called the Euler–Mascheroni constant) is a
mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...
, usually denoted by the lowercase Greek letter
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
(), defined as the limiting difference between the harmonic series and the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
, denoted here by : \begin \gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\ px&=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx. \end Here, represents the
floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
. The numerical value of Euler's constant, to 50 decimal places, is:


History

The constant first appeared in a 1734 paper by the Swiss mathematician
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and for the constant. The Italian mathematician Lorenzo 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. In 1790, he used the notations and for the constant. Other computations were done by Johann von Soldner in 1809, who used the notation . The notation 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 In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
. For example, the German mathematician Carl Anton Bretschneider used the notation in 1835, and
Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the ...
used it in a textbook published in parts from 1836 to 1842. Euler's constant was also studied by the Indian mathematician
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
who published one paper on it in 1917.
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
mentioned the irrationality of as an unsolved problem that seems "unapproachable" and, allegedly, the English mathematician Godfrey Hardy offered to give up his Savilian Chair at
Oxford Oxford () is a City status in the United Kingdom, cathedral city and non-metropolitan district in Oxfordshire, England, of which it is the county town. The city is home to the University of Oxford, the List of oldest universities in continuou ...
to anyone who could prove this.


Appearances

Euler's constant appears frequently in mathematics, especially in
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
. Examples include, among others, the following places: (''where'' '''*' means that this entry contains an explicit equation''):


Analysis

* The Weierstrass product formula for the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
and the Barnes G-function. * The
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
of the gamma function, \Gamma(1/x)\sim x-\gamma. * Evaluations of the
digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...
at rational values. * The
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants. * Values of the derivative of the Riemann zeta function and Dirichlet beta function. * In connection to the
Laplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
and Mellin transform. * In the regularization/
renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
of the harmonic series as a finite value. *Expressions involving the exponential and
logarithmic integral In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
.* * A definition of the cosine integral.* * In relation to
Bessel functions Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
. * Asymptotic expansions of modified Struve functions. * In relation to other
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
.


Number theory

* An inequality for
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
. * The growth rate of the
divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includi ...
. * A formulation of the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
. * The third of Mertens' theorems.* * The calculation of the Meissel–Mertens constant. * Lower bounds to specific prime gaps. * An
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...
of the average number of divisors of all numbers from 1 to a given ''n.'' * The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne primes. * An estimation of the efficiency of the
euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is a ...
. * Sums involving the Möbius and von Mangolt function. * Estimate of the divisor summatory function of the Dirichlet hyperbola method.


In other fields

*In some formulations of Zipf's law. *The answer to the coupon collector's problem.* * The mean of the Gumbel distribution. * An approximation of the
Landau distribution Landau (), officially Landau in der Pfalz (, ), is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990), a lon ...
. * The
information entropy In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed ...
of the Weibull and Lévy distributions, and, implicitly, of the
chi-squared distribution In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
for one or two degrees of freedom. * An upper bound on Shannon entropy in
quantum information theory Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
. * In dimensional regularization of Feynman diagrams in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. * In the BCS equation on the critical temperature in
BCS theory In physics, the Bardeen–Cooper–Schrieffer (BCS) theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory descr ...
of superconductivity.* * Fisher–Orr model for genetics of adaptation in evolutionary biology.


Properties


Irrationality and transcendence

The number has not been proved algebraic or transcendental. In fact, it is not even known whether is
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
. The ubiquity of revealed by the large number of equations below and the fact that has been called the third most important mathematical constant after and makes the irrationality of a major open question in mathematics. However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant and the Gompertz constant is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. Kurt Mahler showed in 1968 that the number \frac \pi 2\frac-\gamma is transcendental, where J_0 and Y_0 are the usual
Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
s. It is known that the
transcendence degree In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients ...
of the field \mathbb Q(e,\gamma,\delta) is at least two. In 2010, M. Ram Murty and N. Saradha showed that at most one of the Euler-Lehmer constants, i. e. the numbers of the form \gamma(a,q) = \lim_\left( - \frac + \sum_^n\right) is algebraic, if and ; this family includes the special case . Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property, where the generalized Euler constant are defined as \gamma(\Omega) = \lim_ \left( \sum_^x \frac - \log x \cdot \lim_ \frac \right), where is a fixed list of prime numbers, 1_\Omega(n) =0 if at least one of the primes in is a prime factor of , and 1_\Omega(n) =1 otherwise. In particular, . Using a
continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
analysis, Papanikolaou showed in 1997 that if is
rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
, its denominator must be greater than 10244663. If is a rational number, then its denominator must be greater than 1015000. Euler's constant is conjectured not to be an algebraic period, but the values of its first 109 decimal digits seem to indicate that it could be a normal number.


Continued fraction

The simple
continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
expansion of Euler's constant is given by: :\gamma=0+\cfrac which has no ''apparent'' pattern. It is known to have at least 16,695,000,000 terms, and it has infinitely many terms
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
is irrational. Numerical evidence suggests that both Euler's constant as well as the constant are among the numbers for which the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
of their simple continued fraction terms converges to
Khinchin's constant In number theory, Khinchin's constant is a mathematical constant related to the simple continued fraction expansions of many real numbers. In particular Aleksandr Yakovlevich Khinchin proved that for almost all real numbers ''x'', the coefficients ...
. Similarly, when p_n/q_n are the convergents of their respective continued fractions, the limit \lim_q_n^ appears to converge to
Lévy's constant In mathematics Lévy's constant (sometimes known as the Khinchin–Lévy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of simple continued fractions. In 1935, the Soviet mathematician Aleksan ...
in both cases. However neither of these limits has been proven. There also exists a generalized continued fraction for Euler's constant. A good simple
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...
of is given by the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of the
square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative nu ...
or about 0.57735: :\frac1\sqrt =0+\cfrac with the difference being about 1 in 7,429.


Formulas and identities


Relation to gamma function

is related to the
digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...
, and hence the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
, when both functions are evaluated at 1. Thus: -\gamma = \Gamma'(1) = \Psi(1). This is equal to the limits: \begin-\gamma &= \lim_\left(\Gamma(z) - \frac1\right) \\&= \lim_\left(\Psi(z) + \frac1\right).\end Further limit results are: \begin \lim_\frac1\left(\frac1 - \frac1\right) &= 2\gamma \\ \lim_\frac1\left(\frac1 - \frac1\right) &= \frac. \end A limit related to the
beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...
(expressed in terms of
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
s) is \begin \gamma &= \lim_\left(\frac - \frac\right) \\ &= \lim\limits_\sum_^m\frac\log\big(\Gamma(k+1)\big). \end


Relation to the zeta function

can also be expressed as an
infinite sum In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
whose terms involve the Riemann zeta function evaluated at positive integers: \begin\gamma &= \sum_^ (-1)^m\frac \\ &= \log\frac4 + \sum_^ (-1)^m\frac.\end The constant \gamma can also be expressed in terms of the sum of the reciprocals of non-trivial zeros \rho of the zeta function: See formula 11 on page 3. Note the typographical error in the numerator of Wolf's sum over zeros, which should be 2 rather than 1. :\gamma = \log 4\pi + \sum_ \frac - 2 Other series related to the zeta function include: \begin \gamma &= \tfrac3- \log 2 - \sum_^\infty (-1)^m\,\frac\big(\zeta(m)-1\big) \\ &= \lim_\left(\frac - \log n + \sum_^n \left(\frac1 - \frac\right)\right) \\ &= \lim_\left(\frac \sum_^\infty \frac \sum_^m \frac1 - n \log 2+ O \left (\frac1\right)\right).\end The error term in the last equation is a rapidly decreasing function of . As a result, the formula is well-suited for efficient computation of the constant to high precision. Other interesting limits equaling Euler's constant are the antisymmetric limit: \begin \gamma &= \lim_\sum_^\infty \left(\frac1-\frac1\right) \\&= \lim_\left(\zeta(s) - \frac\right) \\&= \lim_\frac \end and the following formula, established in 1898 by de la Vallée-Poussin: \gamma = \lim_\frac1\, \sum_^n \left(\left\lceil \frac \right\rceil - \frac\right) where are
ceiling A ceiling is an overhead interior roof that covers the upper limits of a room. It is not generally considered a structural element, but a finished surface concealing the underside of the roof structure or the floor of a story above. Ceilings can ...
brackets. This formula indicates that when taking any positive integer and dividing it by each positive integer less than , the average fraction by which the quotient falls short of the next integer tends to (rather than 0.5) as tends to infinity. Closely related to this is the
rational zeta series In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number ''x'', the r ...
expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit: \gamma =\lim_\left( \sum_^n \frac1 - \log n -\sum_^\infty \frac\right), where is the
Hurwitz zeta function In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by :\zeta(s,a) = \sum_^\infty \frac. This series is absolutely convergent for the given values of and and c ...
. The sum in this equation involves the
harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...
s, . Expanding some of the terms in the Hurwitz zeta function gives: H_n = \log(n) + \gamma + \frac1 - \frac1 + \frac1 - \varepsilon, where can also be expressed as follows where is the
Glaisher–Kinkelin constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to special functions like the -function and the Barnes -function. The constant also appears in a number of sums and i ...
: \gamma =12\,\log(A)-\log(2\pi)+\frac\,\zeta'(2) can also be expressed as follows, which can be proven by expressing the
zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * A ...
as a
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
: \gamma=\lim_\left(-n+\zeta\left(\frac\right)\right)


Relation to triangular numbers

Numerous formulations have been derived that express \gamma in terms of sums and logarithms of
triangular numbers A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
. See formulas 1 and 10. One of the earliest of these is a formula for the
harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...
attributed to
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
where \gamma is related to \textstyle \ln 2T_ in a series that considers the powers of \textstyle \frac (an earlier, less-generalizable proof by
Ernesto Cesàro Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry. He wrote a book, ''Lezioni di geometria intrinseca'' (Naples, 1890), on this topic, in which he also describes ...
gives the first two terms of the series, with an error term): :\begin \gamma &= H_u - \frac \ln 2T_u - \sum_^\frac-\Theta_\,\frac \end From
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...
follows a similar series: :\gamma = \ln 2\pi - \sum_^ \frac The series of inverse triangular numbers also features in the study of the
Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
posed by
Pietro Mengoli Pietro Mengoli (1626, Bologna – June 7, 1686, Bologna) was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647. He remained as professor there ...
. Mengoli proved that \textstyle \sum_^\infty \frac = 1, a result
Jacob Bernoulli Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...
later used to estimate the
value Value or values may refer to: Ethics and social sciences * Value (ethics), concept which may be construed as treating actions themselves as abstract objects, associating value to them ** Axiology, interdisciplinary study of values, including e ...
of \zeta(2), placing it between 1 and \textstyle \sum_^\infty \frac = \sum_^\infty \frac = 2. This identity appears in a formula used by
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
to compute roots of the zeta function, where \gamma is expressed in terms of the sum of roots \rho plus the difference between Boya's expansion and the series of exact
unit fractions A unit fraction is a positive fraction with one as its numerator, 1/. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When a ...
\textstyle \sum_^ \frac: :\gamma - \ln 2 = \ln 2\pi + \sum_ \frac - \sum_^ \frac


Integrals

equals the value of a number of definite
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s: \begin \gamma &= - \int_0^\infty e^ \log x \,dx \\ &= -\int_0^1\log\left(\log\frac 1 x \right) dx \\ &= \int_0^\infty \left(\frac1-\frac1 \right)dx \\ &= \int_0^1\frac \, dx -\int_1^\infty \frac\, dx\\ &= \int_0^1\left(\frac1 + \frac1\right)dx\\ &= \int_0^\infty \left(\frac1-e^\right)\frac,\quad k>0\\ &= 2\int_0^\infty \frac \, dx ,\\ &= \log\frac-\int_0^\infty \frac \, dx ,\\ &= \int_0^1 H_x \, dx, \\ &= \frac+\int_^\log\left(1+\frac\right)dt \\ &= 1-\int_0^1 \ dx \\ &= \frac+\int_^\frac \end where is the fractional harmonic number, and \ is the
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than , called ''floor'' of or \lfloor x\rfloor. Then, the fractional ...
of 1/x. The third formula in the integral list can be proved in the following way: \begin &\int_0^ \left(\frac - \frac \right) dx = \int_0^ \frac dx = \int_0^ \frac \sum_^ \frac dx \\ pt&= \int_0^ \sum_^ \frac dx = \sum_^ \int_0^ \frac dx = \sum_^ \frac \int_0^ \frac dx \\ pt&= \sum_^ \frac m!\zeta(m+1) = \sum_^ \frac\zeta(m+1) = \sum_^ \frac \sum_^\frac = \sum_^ \sum_^ \frac\frac \\ pt&= \sum_^ \sum_^ \frac\frac = \sum_^ \left frac - \log\left(1+\frac\right)\right = \gamma \end The integral on the second line of the equation is the definition of the Riemann zeta function, which is . Definite integrals in which appears include: \begin \int_0^\infty e^ \log x \,dx &= -\frac \\ \int_0^\infty e^ \log^2 x \,dx &= \gamma^2 + \frac \\ \int_0^\infty \frac \,dx &= \frac12 \log^2 2 - \gamma \end We also have Catalan's 1875 integral \gamma = \int_0^1 \left(\frac1\sum_^\infty x^\right)\,dx. One can express using a special case of Hadjicostas's formula as a
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the Real line, r ...
with equivalent series: \begin \gamma &= \int_0^1 \int_0^1 \frac\,dx\,dy \\ &= \sum_^\infty \left(\frac 1 n -\log\frac n \right). \end An interesting comparison by Sondow is the double integral and alternating series \begin \log\frac 4 \pi &= \int_0^1 \int_0^1 \frac \,dx\,dy \\ &= \sum_^\infty \left((-1)^\left(\frac 1 n -\log\frac n \right)\right). \end It shows that may be thought of as an "alternating Euler constant". The two constants are also related by the pair of series \begin \gamma &= \sum_^\infty \frac \\ \log\frac4 &= \sum_^\infty \frac , \end where and are the number of 1s and 0s, respectively, in the base 2 expansion of .


Series expansions

In general, \gamma = \lim_\left(\frac+\frac+\frac + \ldots + \frac - \log(n+\alpha) \right) \equiv \lim_ \gamma_n(\alpha) for any . However, the rate of convergence of this expansion depends significantly on . In particular, exhibits much more rapid convergence than the conventional expansion . This is because \frac < \gamma_n(0) - \gamma < \frac, while \frac < \gamma_n(1/2) - \gamma < \frac. 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 In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
approaches : \gamma = \sum_^\infty \left(\frac 1 k - \log\left(1+\frac 1 k \right)\right). The series for is equivalent to a series Nielsen found in 1897: \gamma = 1 - \sum_^\infty (-1)^k\frac. In 1910, Vacca found the closely related series \begin \gamma & = \sum_^\infty (-1)^k\frac k \\ pt& = \tfrac12-\tfrac13 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right) + 3\left(\tfrac18 - \tfrac19 + \tfrac1 - \tfrac1 + \cdots - \tfrac1\right) + \cdots, \end where is the logarithm to base 2 and is the
floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
. This can be generalized to: \gamma= \sum_^\infty \frac \varepsilon(k)where:\varepsilon(k)= \begin B-1, &\text B\mid n \\ -1, &\textB\nmid n \end In 1926 Vacca found a second series: \begin \gamma + \zeta(2) & = \sum_^\infty \left( \frac1 - \frac1\right) \\ pt& = \sum_^\infty \frac \\ pt&= \frac12 + \frac23 + \frac1\sum_^ \frac + \frac1\sum_^ \frac + \cdots \end From the
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expansion for the logarithm of the gamma function we get: \gamma = \log\pi - 4\log\left(\Gamma(\tfrac34)\right) + \frac 4 \pi \sum_^\infty (-1)^\frac. Ramanujan, in his lost notebook gave a series that approaches : \gamma = \log 2 - \sum_^ \sum_^ \frac An important expansion for Euler's constant is due to
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and
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\gamma = \sum_^\infty \frac = \frac12 + \frac1 + \frac1 + \frac + \frac3 + \cdots, where are
Gregory coefficients Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,Ch. Jordan. ''The Calculus of Finite Differences'' Chelsea Publishing Company, USA, 1947.L. Comtet. ''Adva ...
. This series is the special case of the expansions \begin \gamma &= H_ - \log k + \sum_^\frac && \\ &= H_ - \log k + \frac1 + \frac1 + \frac1 + \frac + \cdots && \end convergent for A similar series with the Cauchy numbers of the second kind is \gamma = 1 - \sum_^\infty \frac =1- \frac -\frac - \frac - \frac - \frac - \ldots Blagouchine (2018) found a generalisation of the Fontana–Mascheroni series \gamma=\sum_^\infty\frac\Big\, \quad a>-1 where are the
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, which are defined by the generating function \frac= \sum_^\infty z^n \psi_n(s) ,\qquad , z, <1. For any rational this series contains rational terms only. For example, at , it becomes \gamma=\frac - \frac - \frac - \frac - \frac - \frac - \frac - \ldots Other series with the same polynomials include these examples: \gamma= -\log(a+1) - \sum_^\infty\frac,\qquad \Re(a)>-1 and \gamma= -\frac \left\,\qquad \Re(a)>-1 where is the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
. A series related to the Akiyama–Tanigawa algorithm is \gamma= \log(2\pi) - 2 - 2 \sum_^\infty\frac= \log(2\pi) - 2 + \frac + \frac+ \frac + \frac+ \frac + \ldots where are the
Gregory coefficients Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,Ch. Jordan. ''The Calculus of Finite Differences'' Chelsea Publishing Company, USA, 1947.L. Comtet. ''Adva ...
of the second order. As a series of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s: \gamma = \lim_\left(\log n - \sum_\frac\right).


Asymptotic expansions

equals the following asymptotic formulas (where is the th
harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...
): *\gamma \sim H_n - \log n - \frac1 + \frac1 - \frac1 + \cdots (''Euler'') *\gamma \sim H_n - \log\left(\right) (''Negoi'') *\gamma \sim H_n - \frac - \frac1 + \frac1 - \cdots ('' Cesàro'') The third formula is also called the Ramanujan expansion. Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations. He showed that (Theorem A.1): \begin \sum_^\infty \Big(\log n +\gamma - H_n + \frac\Big) &= \frac \\ \sum_^\infty \Big(\log \sqrt +\gamma - H_n \Big) &= \frac-\gamma \\ \sum_^\infty (-1)^n\Big(\log n +\gamma - H_n\Big) &= \frac \end


Exponential

The constant is important in number theory. Its numerical value is: equals the following
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, where is the th
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
: e^\gamma = \lim_\frac1 \prod_^n \frac. This restates the third of Mertens' theorems. We further have the following product involving the three constants , and : \frac=\lim_ \log p_n \prod_^n \frac. Other
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound ...
s relating to include: \begin \frac &= \prod_^\infty e^\left(1+\frac1\right)^n \\ \frac &= \prod_^\infty e^\left(1+\frac2\right)^n. \end These products result from the Barnes -function. In addition, e^ = \sqrt \cdot \sqrt \cdot \sqrt \cdot \sqrt \cdots where the th factor is the th root of \prod_^n (k+1)^. This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using
hypergeometric function In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
s. It also holds that \frac=\prod_^\infty\left(e^\left(1+\frac+\frac\right)\right).


Published digits


Generalizations


Stieltjes constants

''Euler's generalized constants'' are given by \gamma_\alpha = \lim_\left(\sum_^n \frac1 - \int_1^n \frac1\,dx\right) for , with as the special case . Extending for gives: \gamma_ = \zeta(\alpha) - \frac1 with again the limit: \gamma = \lim_\left(\zeta(a) - \frac1\right) This can be further generalized to c_f = \lim_\left(\sum_^n f(k) - \int_1^n f(x)\,dx\right) for some arbitrary decreasing function . Setting f_n(x) = \frac gives rise to the Stieltjes constants \gamma_n, that occur in the
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
expansion of the Riemann zeta function: : \zeta(1+s)=\frac+\sum_^\infty \frac \gamma_n s^n. with \gamma_0 = \gamma = 0.577\dots


Euler-Lehmer constants

''Euler–Lehmer constants'' are given by summation of inverses of numbers in a common modulo class: \gamma(a,q) = \lim_\left (\sum_ \frac1-\frac\right). The basic properties are \begin &\gamma(0,q) = \frac, \\ &\sum_^ \gamma(a,q)=\gamma, \\ &q\gamma(a,q) = \gamma-\sum_^e^\log\left(1-e^\right), \end and if the
greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...
then q\gamma(a,q) = \frac\gamma\left(\frac,\frac\right)-\log d.


Masser-Gramain constant

A two-dimensional generalization of Euler's constant is the
Masser-Gramain constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
. It is defined as the following limiting difference: :\delta = \lim_ \left( -\log n + \sum_^n \frac \right) where r_k is the smallest radius of a disk in the complex plane containing at least k
Gaussian integers In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
. The following bounds have been established: 1.819776 < \delta < 1.819833.


See also

* Harmonic series * Riemann zeta function * Stieltjes constants * Meissel-Mertens constant


References

* * *


Footnotes


Further reading

* Derives as sums over Riemann zeta functions. * * * * * * Julian Havil (2003): ''GAMMA: Exploring Euler's Constant'', Princeton University Press, ISBN 978-0-69114133-6. * * * * * * * with an Appendix b
Sergey Zlobin


External links

* *
Jonathan Sondow.


E.A. Karatsuba (2005) * Further formulae which make use of the constant

{{DEFAULTSORT:Euler's constant Mathematical constants Unsolved problems in number theory Leonhard Euler