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 :
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 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,
.
* 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 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.*
* A definition of the
cosine integral.*
* In relation to
Bessel functions.
* Asymptotic expansions of modified
Struve functions.
* In relation to other
special functions.
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 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.
* 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.
* 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 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
is transcendental, where
and
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 of the field
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
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
where is a fixed list of prime numbers,
if at least one of the primes in is a prime factor of , and
otherwise. In particular, .
Using a
continued fraction 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 10
244663. If is a rational number, then its denominator must be greater than 10
15000.
Euler's constant is conjectured not to be an
algebraic period, but the values of its first 10
9 decimal digits seem to indicate that it could be a
normal number.
Continued fraction
The simple
continued fraction expansion of Euler's constant is given by:
:
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. Similarly, when
are the convergents of their respective continued fractions, the limit
appears to converge to
Lévy's constant 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 of is given by the
reciprocal 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:
:
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:
This is equal to the limits:
Further limit results are:
A limit related to the
beta function (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
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:
The constant
can also be expressed in terms of the sum of the reciprocals of
non-trivial zeros 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.]
:
Other series related to the zeta function include:
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:
and the following formula, established in 1898 by
de la Vallée-Poussin:
where are
ceiling 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 expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:
where is the
Hurwitz zeta function. 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:
where
can also be expressed as follows where is the
Glaisher–Kinkelin constant:
can also be expressed as follows, which can be proven by expressing the
zeta function 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 ...
:
Relation to triangular numbers
Numerous formulations have been derived that express
in terms of sums and logarithms of
triangular numbers.
[ 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 where
is related to
in a series that considers the powers of
(an earlier, less-generalizable proof by
Ernesto Cesàro gives the first two terms of the series, with an error term):
:
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:
:
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. Mengoli proved that
, 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 of
, placing it between
and
. 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
is expressed in terms of the sum of roots
plus the difference between Boya's expansion and the series of exact
unit fractions :
:
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:
where is the
fractional harmonic number, and
is the
fractional part of
.
The third formula in the integral list can be proved in the following way:
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:
We also have
Catalan's 1875 integral
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:
An interesting comparison by Sondow is the double integral and alternating series
It shows that may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series
where and are the number of 1s and 0s, respectively, in the
base 2 expansion of .
Series expansions
In general,
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
while
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 :
The series for is equivalent to a series
Nielsen found in 1897:
In 1910,
Vacca found the closely related series
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:
where:
In 1926 Vacca found a second series:
From the
Malmsten–
Kummer expansion for the logarithm of the gamma function
we get:
Ramanujan, in his
lost notebook gave a series that approaches :
An important expansion for Euler's constant is due to
Fontana and
Mascheroni
where are
Gregory coefficients. This series is the special case of the expansions
convergent for
A similar series with the Cauchy numbers of the second kind is
Blagouchine (2018) found a generalisation of the Fontana–Mascheroni series
where are the
Bernoulli polynomials of the second kind, which are defined by the generating function
For any rational this series contains rational terms only. For example, at , it becomes
Other series with the same polynomials include these examples:
and
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
where are the
Gregory coefficients 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:
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 ...
):
*
(''Euler'')
*
(''Negoi'')
*
(''
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):
Exponential
The constant is important in number theory. Its numerical value is:
equals the following
limit, 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 ...
:
This restates the third of
Mertens' theorems.
We further have the following product involving the three constants , and :
Other
infinite products relating to include:
These products result from the
Barnes -function.
In addition,
where the th factor is the th root of
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
Published digits
Generalizations
Stieltjes constants
''Euler's generalized constants'' are given by
for , with as the special case . Extending for gives:
with again the limit:
This can be further generalized to
for some arbitrary decreasing function . Setting
gives rise to the
Stieltjes constants , 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:
:
with
Euler-Lehmer constants
''Euler–Lehmer constants'' are given by summation of inverses of numbers in a common
modulo class:
The basic properties are
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
Masser-Gramain constant
A two-dimensional generalization of Euler's constant is the
Masser-Gramain constant. It is defined as the following limiting difference:
:
where
is the smallest radius of a disk in the complex plane containing at least
Gaussian integers.
The following bounds have been established:
.
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